What replaces it, though, was anything but clear.
The newly powerful in France and Greece want to roll back the spending cuts and tax increases that have defined Europe's response to its 3-year-old debt crisis. But campaign rhetoric is likely to prove more extreme than any real-world reversal of the budget tightening.
World financial markets took Europe's latest round of political upheaval in stride, convulsing early and then recovering. The continent's uncertain future — including the possibility of Greece leaving the euro — was causing anxiety but not panic about the threat to the global economy.
But there is hardly unity in Europe.
Sunday night, Socialist president-elect Hollande celebrated his victory over Nicolas Sarkozy by vowing, "Austerity can no longer be inevitable!"
On Monday, German Chancellor Angela Merkel gently pushed back.
She rejected Hollande's call to renegotiate a treaty signed last month on tougher action to control government deficits. "We in Germany, and I personally," she said, "believe the fiscal pact is not up for negotiation."
Still, she stressed the importance of French-German cooperation — and her willingness to meet soon with Hollande.
Economists said that while the anti-austerity winds are bound to stir up short-term political instability, especially in Greece, they could eventually bring some financial calm.
"This is going to force some rethinking" all across Europe about how to manage the debt crisis, said Laura Gonzalez, a finance professor at Fordham University in New York. "That is good for everybody."
Greece remains the focus of Europe's financial and political unease. Political parties that made gains by rejecting belt-tightening still have to assemble a majority coalition in Parliament before they can begin governing. The conservatives got the first try Monday but failed — leaving a new left-wing, party to take its turn. If no party can assemble a coalition, the country will need to hold new elections, probably in June.
The main stock index in Greece plunged almost 7 percent. France's CAC-40 ended 1.7 percent higher.
The biggest fear was that Greece's new leadership would renege on commitments made to secure the country's massive rescue loans — or even leave the euro. Merkel pressed Greek leaders on Monday to stay the course. "Of course, the most important thing is that the programs we agreed with Greece are continued," she said.
Greece wasn't the only problem. The 17 countries that use the euro — and nine other European countries — agreed in March to the fiscal compact that seeks to make countries balance their budgets. But bailout fears have intensified in recent months as Spain, Italy and other governments face rising borrowing costs on bond markets, a sign that investors are nervous about the size of their debts relative to their economic output. Austerity was intended to address these jitters by reducing their government's borrowing needs, but there has been a negative side effect: As economic output shrinks, the debt burden actually looks worse.
As Europe's economy got weaker, the public and politicians grew weary of the budget-cutting required to make the fiscal compact work. Across Europe, austerity meant layoffs and pay cuts for state workers, scaled-back expenditures on welfare and social programs, and higher taxes and fees to boost government revenue.
Hollande says he intends to renegotiate the fiscal treaty so that it places an emphasis on growth and not just deficit reduction. He says governments should actually increase spending now, while economies are so weak.
Merkel and the European Central Bank have instead stressed deeper, long term fixes such as reducing red tape for small businesses, making it easier for workers to find jobs across the eurozone and breaking down barriers that countries have created to protect their own industries. Those changes involve challenging unions and other powerful constituencies — and they can take years to have an effect.
The anti-austerity sentiment appears to be picking up strength.
In Italy on Monday, several candidates in local elections who oppose the deficit-cutting promoted by Premier Mario Monti had a strong showing. And the head of the International Monetary Fund — one of the institutions that designed the Greek bailout and the austerity measures that go with it — warned that Europe has to be careful about pushing austerity too far. Christine Lagarde said European countries should reduce their budget deficits gradually to avoid further damage to their economies.
Eight of the 17 eurozone nations are already in recession and unemployment across the bloc rose to 10.9 percent in March — its highest ever.
The European commission called upon Greece to make "full and timely" implementation of its budget cuts. Those include €11.5 billion in new cutbacks that must be found in June to make sure Greece keeps getting money under the terms of its second, €130 billion bailout.
Economist Christoph Weil at Commerzbank said that if aid is cut off, Greece would be unable to pay its debts by autumn, leading to a second default following a writedown of €107 billion in March.
"There is certainly room for negotiations that could save face for both sides," Weil said. But "patience is wearing thin" with Greece and "I would not count on this happening."
The U.S. and European financial systems are so intertwined that a loss of confidence in Europe could cloud the U.S. economy.
But there were few expectations that the votes in France and Greece would have much immediate effect in the U.S.
Still, they make it more likely that Greece will have to abandon the euro, roiling world markets, said Jacob Kirkegaard, research fellow at the Peterson Institute for International Economics.
Sarkozy and Merkel were the architects of the European austerity plan — so close they were known as "Merkozy." The big question now is if there will be a "Merkollande" in Europe's future.
Hollande's plans to jump-start the French economy by investing in infrastructure and buoying small businesses will determine how bumpy the road ahead is.
He has promised to keep the deficit in check by raising taxes on the wealthy and closing some corporate loopholes — but some investors say that will kill the very growth he hopes to foster.
If he does start wildly increasing spending, France will no doubt see its borrowing costs rise — which could make his policies untenable and prompt a shift back to austerity. It was those rising borrowing costs that eventually forced fellow eurozone nations Greece, Ireland and Portugal to seek bailouts.
Some are hoping that Hollande will turn out to be more pragmatic.
"Adieu, election campaign. Bonjour, reality," read an editorial in Germany's daily Sueddeutsche Zeitung.