When Christine Lagarde became the first female finance minister of a major global economy, it's a measure of how much happier the world was back then that media interest focused chiefly on her talent for synchronised swimming. She was the foxy Frenchwoman who'd won medals in the national team in her teens, then worked her way up to chair an American law firm in Chicago, before being invited back to Paris in 2005 as trade minister and promoted two years later to the Treasury. Journalists had lots of fun picturing her upside down in a pool, wearing waterproof lipstick and a nose clip – and Lagarde played along with the joke, crediting the sport with teaching her a useful political skill: "To grit your teeth and smile."

No one is writing about synchronised swimming any more. On the day we met last week, the papers were agog with economic Armageddon, as the new French president flew off to Berlin to face a German chancellor whose austerity creed appeared to be on a collision course with France's new mission for growth. Athens was unravelling into chaos, unable to form a government and forced into fresh elections, plunging the markets into freefall as Europe's leaders abandoned any pretence that a Greek exit from the euro might not be imminent. The future of the euro itself was, one headline declared, "a chronicle of a death foretold". When François Hollande's plane was struck by lightning, the heavens themselves seemed to be trying to tell us just how much trouble we are in.

Coming face to face with Lagarde, however, you could be forgiven for thinking you must have imagined the whole crisis. We meet at the Paris office of the International Monetary Fund (IMF), a concrete grey modernist building so unassuming as to lack even a sign advertising its existence. Inside, the decor is plain and functional, the atmosphere eerily hushed. An empty lift glides up to a floor of deserted offices, where I wait by myself for a while until a tall, strikingly self-possessed woman appears and greets me with the elegant serenity of a Parisian hostess receiving a dinner party guest. "Let us sit here," she suggests, ushering me to a window seat beside a vase of flowers. "You can look at my orchids."

The managing director of the IMF may look like one of those statuesque silvery models who appear in Weekend's All Ages fashion pages, but she is one of the world's most powerful women, in the eye of the world's worst storm in living memory. In the years leading up to the 2008 crash, the IMF had been starting to look, if not quite redundant, then not massively important; most of the world's economies appeared to be ticking along quite happily without it. But the crash changed everything, so I'm curious to know when she first thought of running for the job. Actually, she says, it wasn't her idea, but George Osborne's.

"We were travelling together and we were sort of thinking about the political scene, and he said you know [Dominique] Strauss-Kahn is bound to be a candidate for the French presidential elections. What's going to happen with the IMF? Have you thought about it? That's how it started. That's when I started to play with the idea."

But events moved faster than expected last May – "Yes, faster than we ever thought!" – when the incumbent, Strauss-Kahn, was accused of the attempted rape of a New York hotel... and forced to resign. On top of the sex scandal there was a ding-dong over whether the post should go, as it always has, to another European – another French one, at that – when the global economy today bears no resemblance to the one for which the job was originally designed in 1945. A French candidate would have to be extraordinarily impressive – and Lagarde is certainly that, lauded by everyone from Alistair Darling to Timothy Geithner, who praised her "lightning-quick wit, genuine warmth and ability to bridge divides". But, 67 years after its creation, I'm not sure everyone even really understands exactly what the head of the IMF is meant to do, so I ask Lagarde to explain in words an 11-year-old would understand.

"Well, I look under the skin of countries' economies and I help them make better decisions and be stronger, to prosper and create employment." You could think of the IMF as a global payday loan company for countries who have got into trouble and can't meet their financial commitments – the difference being that instead of charging sky-high interest rates, it demands radical economic reforms. And if they say they don't like the sound of that? "If I'm confident that the sound of it is accurate, I say, well, I'm terribly sorry but this is the sound we are making."

Voters in Greece and France have decided they don't like the sound of it at all and so, as the crisis accelerates, Lagarde's job is looking increasingly indivisible from a mission to save the euro. Some critics have suggested that the appointment of a Europhile former French finance minister was akin to putting a drunk behind the bar; a former IMF chief economist has warned she is essentially in denial about the fundamental flaws of the euro and likely to "throw loans" at its problems, while Ed Balls has argued, "The IMF's job is to support individual countries with solvency crises, not to support a whole monetary union which cannot agree the necessary steps to maintain itself." So I ask if she would be trying just as hard to save the single currency if she were, say, Mexican.

"Yeah yeah yeah yeah yeah. There is an emotional side of me that is pro-European," she acknowledges. "But I try to not be French, not be European, when I do my job. And I know that resolving the Euro area crisis matters also to the Mexican, the Australian, the Brazilian."

She has travelled the world asking countries to contribute to a firewall fund, but several have asked – not unreasonably – why they should have to pay for Europe's mistakes, when Europe is still richer than most of the world. Does the eurozone crisis matter more to their own interests than they realise? "Oh, I think they realise it," Lagarde says quickly, sounding deadly serious. "There has not been a capital I have visited in the last 10 months where the first question has not been: what is the situation in Europe? Are the Europeans sorting it out?"

Nevertheless, while this might come as a surprise to Greeks suffering under extreme austerity, some say Lagarde's approach to the eurozone is less draconian than the IMF's traditional policy towards developing world economies. Is it easier to impose harsh demands upon small economies, but much harder to tell difficult truths to the big ones – particularly fellow Europeans? "No," she says firmly. "No, it's not harder. No. Because it's the mission of the fund, and it's my job to say the truth, whoever it is across the table. And I tell you something: it's sometimes harder to tell the government of low-income countries, where people live on $3,000, $4,000 or $5,000 per capita per year, to actually strengthen the budget and reduce the deficit. Because I know what it means in terms of welfare programmes and support for the poor. It has much bigger ramifications."

So when she studies the Greek balance sheet and demands measures she knows may mean women won't have access to a midwife when they give birth, and patients won't get life-saving drugs, and the elderly will die alone for lack of care – does she block all of that out and just look at the sums?

"No, I think more of the little kids from a school in a little village in Niger who get teaching two hours a day, sharing one chair for three of them, and who are very keen to get an education. I have them in my mind all the time. Because I think they need even more help than the people in Athens." She breaks off for a pointedly meaningful pause, before leaning forward.

"Do you know what? As far as Athens is concerned, I also think about all those people who are trying to escape tax all the time. All these people in Greece who are trying to escape tax."

Even more than she thinks about all those now struggling to survive without jobs or public services? "I think of them equally. And I think they should also help themselves collectively." How? "By all paying their tax. Yeah."

It sounds as if she's essentially saying to the Greeks and others in Europe, you've had a nice time and now it's payback time.

"That's right." She nods calmly. "Yeah."

And what about their children, who can't conceivably be held responsible? "Well, hey, parents are responsible, right? So parents have to pay their tax."

Lagarde is a beguiling mixture of steel and silk, for she can switch seamlessly from this sort of hardball talk to nimble diplomacy. Asked if she expects to be the last European to run the IMF, she replies, "Well, I hope I'm not the last woman." But the last European? "I don't know." She smiles, adding playfully, "I might last for a long time."

I begin a question about British Eurosceptics – "Lots of people where I come from – " but she can see what's coming and interjects warmly, "A beautiful island." When I ask if she enjoyed dealing with Gordon Brown, she offers, "Erm… I don't think he was ever finance minister when I was." That's a rather graceful way of avoiding the question, I say, smiling. Lagarde affects a blank expression of innocence, and starts to laugh.

Everybody talks about Lagarde's phenomenal charm and it doesn't take long in her company to see why. She goes, "Pouff!" when I say so, batting the compliment away with a flick of the wrist, but she is neither an economist, nor even really a politician – she spent just six years of her career as a minister in France – so I wonder if charm is actually the key qualification for the job.

"Well, I think when you drill down and ask what it takes to be managing director of the IMF, then the ability to listen, the ability to understand the perspective of your entire membership, the respect and tolerance for the political diversity, the cultural diversity, I think that's very important actually. I mean, it's often underestimated because many people will say you need to be a very strong economist. Well, maybe so. But I wouldn't qualify for the job. I'm not the top-notch economist; I can understand what people talk about, I have enough common sense for that, and I've studied a bit of economics, but I'm not a super-duper economist. But, yes, that appreciation for the interests pursued by the other side at a negotiation table, a sense of the collective interest and how that can transcend the vested individual interests of the members, that matters."

She doesn't claim these as feminine virtues, but acknowledges, "I've criticised enough women who are fighting so hard to look like a man that it destroys half of their own sanity and humanity." How often does she feel judged as a woman at the IMF? "Oh, quite often. That wouldn't surprise you! Come on." When she had the temerity last autumn to point out the obvious truth that Europe's banks were under-capitalised, "that's an occasion where I think some observations were related to me being a woman." She drops her voice to mimic the catty whispers: "'She doesn't know what she's talking about, silly woman, she must have been poorly advised.'"

So what does she do about it? "I think you can choose one of two options. Either you become bitter, and you complain constantly about it, and argue that people will criticise you or undermine you because you are a woman. Or you decide to take advantage of it. Not overplaying the feminine side of things; not being on the seducing side, not playing the attractive woman in high heels – I've never done that and I think my mother would be horrified if I did, and I don't want that to happen because I loved her very much. But…" She falls silent. But what? "Men will not insult you or will not easily say no when you tell them you need more money to secure the institution and make sure it can do its work." Does she mean it's easier for her to ask for money as a woman? "Yes," she flashes back. "Yes. Yes. Absolutely." Because masculinity responds to a woman saying I need more money? "Yes," she agrees, smiling. "People have said that to me. 'How can I say no to you?'"

For all Lagarde's charm, it's hard not to feel a sense of Alice In Wonderland bewilderment about the IMF's work. The Americans are recovering with a stimulus programme more familiar to Europe than Washington, while a Frenchwoman is trying to save the eurozone with austerity measures that would please the Tea Party. The whole point of the European project was to prevent the sort of conflict that once engulfed the continent, and yet the IMF's life support strategy has seen neo-Nazis elected in Athens, and now risks destabilising the marriage between Germany and France on which the European dream depends. When democratic elections produce politicians unwilling to play by the IMF's rules, they have been replaced by unelected technocrats – Mario Monti in Italy, Lucas Papademos in Greece – gifting Eurosceptics evidence for their charge that the EU is fundamentally anti-democratic.

Were voters in Greece and France basically wrong to elect anti-austerity politicians? "You are never wrong when you have voted because you've acted in accordance with your conscience and your beliefs, and you've exercised your democratic right, which is, you know, perfectly legitimate in our democracies."

But Germans elected Hitler in 1933, and we don't think they were right, do we?

"Well, somebody once said if people are not happy with their government, you change the people." She laughs, deftly sidestepping the question. "What's really interesting," she says more seriously, "is that wherever you see a change of government, for instance in Spain, do you see major changes from the economic and financial policies that were conducted by their predecessors? No." She suspects we will see a similar pattern now in France. "I'm very much a believer that it's action that matters much more so than, you know, the flurry of political promises and statements and slogans that are used during political campaigns. So let's see."

Is she saying there's no need to panic about a rift between Paris and Berlin? "I should think so," she agrees quietly, with a knowing smile. "I think it's largely overstated."

Lagarde's unflappable calm seems to come quite naturally. She was born in Paris in 1956, the eldest daughter of a university lecturer and a teacher; her father suffered from motor neurone disease and died when she was just 17. After failing twice to get into the prestigious Ecole Nationale d'Administration, the elite incubator for French civil servants, she joined the American law firm Baker & McKenzie and rose to become its first female chairman. In her early 30s she had two sons with her first husband, but after that the details get a little hazy; she married again while in Chicago, to a British businessman, but now lives with a Corsican she first met in her 20s at law school. In the French tradition, that's about as much as we know of her private life, apart from the fact that she is teetotal, vegetarian and a fanatical swimmer who will stay only in hotels that have pools. "She radiates," an acquaintance once said of her. "I think that's because she swims so much."

She certainly radiates assurance, but of course part of being reassuring means not saying anything very bold. I ask how she squares austerity with growth, but she thinks the furious debate between the two is generating more heat than light.

"What we say is it cannot be either or; it's not either austerity or growth, that's just a false debate. Nobody could argue against growth. And no one could argue against having to repay your debts. The question and the difficulty is how do you reconcile the two, and in which order do you take them? I would argue that you do it on a country by country case; it's not going to be a one size fits all."

In the UK's case, Lagarde thinks we are broadly on the right lines; public spending cuts, quantitative easing and low interest rates all meet with her approval. I tell her it doesn't feel that way to a lot of people here, and ask for an exit narrative – the story of how we'll get out of this mess – that could cheer people up.

"There will be an exit," she says firmly. "No question about it." Yes, but what is it? "Well, we're going to invent it. To give you a couple of positive messages, firstly, protectionism is not reappearing. The second reason for optimism is, there's a lovely sentence by Robert Musil, which says, 'Man is capable of anything – including the best.' And when you see how a situation can be turned around by one individual – get Mr Berlusconi out, you bring Mr Monti in, he's dedicated, he couldn't care less about his political future because he's not interested. And he does the job. And he changes the perception, and restores confidence. That's also a sign of hope."

That may be true, but it's not an exit strategy narrative. "Ah," she says briskly, laughing. "You'll have to come back for that." Could she at least say where we are on the curve; is this as bad as things can be, or will they get worse? "I'm not in the business of reading tea leaves. I don't have a crystal ball. Some of the major issues are being resolved – but it's not over now. Let's face it, it's not over yet."

And for Greece – is the euro over? Lagarde won't say. I ask if I'll be packing euros if I go on holiday to Greece next year and she just smiles. "A holiday in Greece, it's a good investment for the country!"

She will put her name to just one firm prediction: she's going to be at the synchronised swimming at the Olympics this summer. "Osborne promised me that I would be invited, so, yes, I will try to do that. I'm desperate to."

I try once more. When history books are written about the financial crisis, they will say it began in 2008. What date will they give for its end? "Hmm, after the hyphen? After 2008? Two thousand," she says firmly. How odd, I think – her English is perfect, but she must have misunderstood the question. I ask again, but she is laughing. There was no misunderstanding.

"Well, I'm sure about the first two digits: 20. But I'm not sure about the last two digits."