BERLIN — Germany’s center-left opposition won a wafer-thin victory over Chancellor Angela Merkel’s coalition in a major state election Sunday, dealing a setback as she seeks a third term at the helm of Europe’s biggest economy later this year.

The opposition Social Democrats and Greens won a single-seat majority in the state legislature in Lower Saxony, ousting the coalition of Merkel’s conservative Christian Democratic Union and the pro-market Free Democrats that has run the northwestern region for 10 years. The same parties form the national government.

The 58-year-old Merkel will seek another four-year term in a national parliamentary election expected in September. She and her party are riding high in national polls, but the opposition hoped the Lower Saxony vote would show she is vulnerable.

The outcome could boost what so far has been a sputtering campaign by Merkel’s Social Democratic challenger, Peer Steinbrueck.

“This evening gives us real tailwind for the national election,” said Katrin Goering-Eckardt, a leader of Steinbrueck’s allies, the Greens. “We can and will manage to replace the (center-right) coalition.”

However, the close outcome also underscores the possibility of a messy result in September, with no clear winner.

Before Sunday’s election, the main question had been whether the Free Democrats, whose support has eroded badly since they joined Merkel’s national government in 2009, would win the 5 percent needed to gain seats in the state legislature. Polls over recent months had suggested that they might not.

The Free Democrats won 9.9 percent of the vote, thanks to tactical voting by supporters of Merkel’s conservatives. Many chose the smaller party so that the coalition’s “good policies for solid budgets, safe jobs and good education could be continued,” said the general secretary of Merkel’s party, Hermann Groehe.

Despite the popularity of David McAllister, the state’s CDU governor, that helped push down the conservatives’ support to 36 percent from 42.5 percent in Lower Saxony’s last election in 2008. They finished as the biggest single party but fell short of expectations.

The combination of the CDU and Free Democrats has now lost control of four states since the smaller party joined Merkel’s government in 2009.

At national level, the alliance has developed a poor image, with the Free Democrats taking much of the blame for frequent government infighting. Merkel and her party, meanwhile, have been bolstered by a relatively robust economy, low unemployment and the chancellor’s hard-nosed handling of Europe’s debt crisis.

Sunday’s vote may calm intense recent speculation over whether the Free Democrats will force out their embattled leader, Vice Chancellor Philipp Roesler, whose home state is Lower Saxony.

Roesler proclaimed Sunday “a great day” for his party.

Merkel also has profited so far from stumbles by Steinbrueck, a former finance minister who drew criticism in recent weeks for saying that the chancellor earns too little — adding to the controversy over his own high earnings from speeches.

His party improved a little on its feeble performance five years ago in Lower Saxony, polling 32.6 percent. Its allies, the Greens, made bigger gains, to 13.7 percent.  That gave them 49 and 20 seats, respectively, in the state legislature in Hannover; the CDU took 54 and the Free Democrats 14.

Opposition leaders argued that their overall gains and the losses for the center-right showed that a change to a center-left government is possible in Berlin, despite unpromising recent polls.  

Steinbrueck conceded that his recent troubles hadn’t helped, though.

“I am well aware that there was no tailwind from Berlin, and I am also aware that I bear a certain share of responsibility for that,” he told supporters. He said he felt “a certain amount of relief” at the result.

Two parties that have drained support from the center-left over recent years, the Left Party and the Pirate Party, failed to get enough votes Sunday to enter Lower Saxony’s legislature.

A change of government in Lower Saxony will give opposition parties a majority in Parliament’s upper house, which represents Germany’s 16 states.

Though that has limited practical effect, it would enable the Social Democrats and Greens to showcase their plans for Germany by sending policy initiatives to the lower house — controlled by Merkel’s coalition — on issues such as their call for a mandatory national minimum wage.

However, it remains to be seen whether they will be able to build up momentum and prevent Merkel keeping power — possibly by finding a new coalition partner.

With eight months left before the national vote, “so much can change that I’d just say we should be careful about projecting such results onto the national level,” Lothar Probst, a political scientist at the University of Bremen, told Phoenix television.