For seven hours, European finance ministers in a windowless room argued over how to keep a government-debt crisis from infecting eurozone economies, and the meeting threatened to fall into disarray.

Then Christine Lagarde stood up to speak.

She read a proposed solution from notes she had jotted while the others were debating, said Thomas Wieser, a senior European civil servant who was there. “A group of ministers alternating between screaming and mumbling, some of them not necessarily well-informed, gradually went quiet.”

The ministers, said Mr. Wieser and other participants, rallied to Ms. Lagarde’s side.

That was May 2010 in Brussels, when Ms. Lagarde was French finance minister. The topic was how to prevent Greece’s financial crisis from threatening the eurozone’s survival. The brand of political compromise Ms. Lagarde displayed there would play out during her tenure as the International Monetary Fund’s head from June 2011. This month, it helped her become the person tapped by European leaders to helm the European Central Bank, the most influential monetary-policy maker after the U.S. Federal Reserve.