The former head of the International Monetary Fund succeeds Mario Draghi, who as head of the central bank for the 19 countries that use the euro helped keep the currency union together through a financial crisis.
While the crisis has abated, the pressures of the job have not diminished, with the ECB president acting as the final backstop for Europe's economy and with global uncertainties on the rise. Here's a look at the main challenges Lagarde faces.
A WEAKENING ECONOMY
Growth data have weakened even since Draghi announced a big stimulus package on Sept. 12. Analysts think Lagarde may not have to change policy for a while as that stimulus runs, but big risks still loom over the global and European economies. Trade disputes between the U.S. and major economies like China and Europe have hurt manufacturing. Brexit has yet to be resolved, keeping businesses uncertain about how to invest.
The ECB's next move may be to provide even more stimulus, rather than raising interest rates back to where they were before the global financial crisis.
CENTRAL BANK POWER
And yet providing more stimulus could be complicated, in part because so much has already been deployed.
The ECB's benchmark interest rate on deposits left overnight from banks is already at an unprecedented minus 0.5%. Bank officials have said it could be cut further, but at some point the ECB would reach a point where adverse side effects such as the impact on bank profits may outweigh the benefits.
Draghi's plan to have the ECB buy 20 billion euros ($22 billion) a month in corporate and government bonds will continue to help hold down market borrowing costs for companies and governments. Eventually, however, there won't be enough government bonds to buy up. The ECB has a self-imposed limit of buying no more than a third of any government's debt.
And it's not clear how much good more rate cuts and bond purchases would do.
Rates are already negative, and the ECB already pumped 2.6 billion euros ($2.9 billion) into the economy through bond purchases from 2015 to the end of 2018. Despite that, inflation has remained below the ECB's goal of under 2%. Growth has not led to increases in wages and inflation as seen in earlier eras, and not just in Europe; economists say aging populations, digitalization and globalization that can shift jobs to lower wage locations may all play a role.
"The workings of the economy are changing," said Maria Demertzis, deputy director of the Bruegel think tank in Brussels. "This is a big challenge for the ECB: what can they do to stimulate the economy, both because they have almost exhausted the instruments that they have, but also because the digital transformation implies that the economy simply works differently."
Precisely that question led to unusually strong criticism from several members on the ECB governing board, including the central bank heads from Germany, the Netherlands and Austria.
Lagarde can use her political and negotiating skills honed as French finance minister and head of the IMF to manage the rift.
Yet it's more than a matter of winning people over with diplomacy. The dispute reflects uncertainty about how the economy works and how it will respond, or not, to stimulus.
Lagarde, an engaging public speaker, will also need all her communications skills to rebut criticism from German economists and news media that the bank's low-rate policies are depriving savers of returns and benefitting shaky southern European governments. Draghi has often noted that 11 million jobs were created since the peak of post-crisis unemployment in 2013.
Lagarde repeated as recently as Wednesday Draghi's plea for governments that are in good financial shape to spend more on projects such as infrastructure that can help economic growth, and to not let the central bank carry the whole burden. Germany, Europe's biggest economy, has insisted on running budget surpluses.
"After the crisis in 2008 everyone worked together and countries ... did their job. But since then ... countries that have budgetary space have not made the necessary efforts" in investment to support growth, Lagarde said on RTL radio. "Notably countries like Netherlands, Germany and some others in the world."
"Those that have a margin of maneuver - why not use this budget surplus and invest in infrastructure, which really needs it? Why not invest in education, why not invest in innovation?"
The broader question, said Demertzis, is that the eurozone lacks a central budget to stabilize its economic ups and downs but instead relies on budget decisions in 19 member countries. French President Emmanuel Macron has pushed for a central budget, but German skepticism has limited its size and purpose.
"It really needs to be a fiscal capacity at the eurozone level," said Demertzis. "Politically, that is a very, very difficult thing to do. If Lagarde has the political acumen, which I think she does, she may be able to move that conversation in that direction, which would be extremely welcome."
Angela Charlton in Paris contributed to this report.