The Alternative for Germany party dominated a crucial state election on Sunday, according to near-final results, but fell short of the absolute majority needed to become the first far-right party to govern part of the country since the Nazi era.

If those results hold, the AfD, as the party is known by its German initials, could still assume power in the eastern state of Saxony-Anhalt in the coming weeks, depending on the negotiations among rival parties that have vowed not to support an AfD government.

Saxony-Anhalt is a relatively small state, and the heart of the AfD’s power base in eastern Germany. But the election result sent shock waves through the country’s political system, which was designed after World War II to prevent extremist parties from ever returning to power.

“The AfD’s electoral victory in Saxony-Anhalt is less a reflection of the appeal of its political agenda than a stark demonstration of the political establishment’s failure to articulate a convincing vision for the state’s economic, political and social future,” said Jana Puglierin, the head of the Berlin branch of the European Council on Foreign Relations.

The political establishment has treated the AfD as a pariah. Still, the party tapped into voter frustration on a wide range of economic and social issues to win about 44 percent of the vote in the state, and was set to claim 39 of the 83 seats in the Parliament.

“We’ve made history, and we’re taking our country back,” Ulrich Siegmund, the party’s candidate to lead the state, told a cheering crowd on Sunday. “It all starts here in Saxony-Anhalt.”

Exit polls suggested that many of the AfD’s gains came at the expense of the center-right Christian Democrats, the party of Chancellor Friedrich Merz that also led the incumbent government in the state. Amid record turnout, the AfD won the support of many Germans who were casting ballots for the first time.

Germany’s domestic intelligence agency has designated the AfD as a “suspected extremist” group and its Saxony-Anhalt branch as “extremist.” Some of the party’s leaders have played down the Holocaust and adopted Nazi slogans. Others have strong ties to Russia. The AfD has also said that Islam is incompatible with the German Constitution and has vowed to conduct mass deportations of immigrants.

The AfD has rejected accusations of extremism as politically motivated. And voters ignored those accusations because, many told pollsters on Sunday, they saw the AfD as providing better answers to economic, security and immigration concerns than the political mainstream.

Thomas Schröder, 63, cycled to his polling station in central Magdeburg to vote for Mr. Siegmund and the AfD. “He gives me hope, so he gets my vote,” Mr. Schröder, a heating technician, said of Mr. Siegmund. He added that he’d lost trust in other parties. “I experimented with almost all of them, and what did it get me?” he said.

The result buoyed AfD fans across the country and around the world. President Trump posted an image of an exit poll showing the AfD lead to his social media platform, Truth Social.

Elias Beck, 70, of Bavaria, hunkered over his phone outside the main train station in Magdeburg, the state’s capital, studying a chart that showed the AfD close to securing an absolute majority in early results. He said he hoped the party would prevail because the German government had failed to address serious problems at home like illegal migration, while spending hundreds of millions of dollars abroad.

“You have all the problems in Germany, but all of the money is going to Ukraine,” he said.

Some rival parties appeared to have galvanized supporters against the AfD, an effort that some leaders in Berlin vowed was just the start of a backlash against a potential AfD government.

Lars Klingbeil, the country’s vice chancellor and a center-left Social Democrat, said on television that the election was “a turning point,” adding that there were many people “across Germany, who are afraid and worried about this election result for the AfD.”

One of the worried was Muhammad Hassan, 27, a native of Pakistan, who said on Sunday night in Magdeburg that he has done all he can to assimilate in Germany since he arrived in 2023.

Mr. Hassan said he had learned German, obtained a driver’s license, finished a master’s in water engineering and secured a job as a technical draftsman drawing underground infrastructure like sewage and electrical lines. But he worried an AfD government would take it away, a year before he is eligible for permanent residency.

“When I see today’s situation,” he said, referring to the election, “it has totally disappointed me because what I have dreamed, I cannot see right now.”

Rival parties could still prevent the AfD from assuming power in Saxony-Anhalt. The Christian Democrats, who finished a distant second, could end up leading the state if they can build a coalition with four others: the center-left Social Democrats and Greens, the far-left Die Linke, and the populist Sahra Wagenknecht alliance, known by its German initials BSW.

But some Christian Democrat leaders in the state have said they would not work with Die Linke, calling it extremist. BSW leaders have also said they would not support the current state leader, Sven Schulze of the Christian Democrats, or the AfD. They have pushed for a nonpartisan state leader, an idea Mr. Siegmund rejected late on Sunday. Mr. Siegmund also seemed to invite some members of the Christian Democrats to defect and supply the votes for an AfD government.

The coming coalition wrangling could take days or weeks. Analysts said it was likely to leave the state’s next leader in fragile condition, no matter which party is in power.

Sunday’s result, said Sudha David-Wilp, the vice president of external relations at the German Marshall Fund, a think tank, “stands as a harbinger for the inability for mainstream parties to form stable coalitions.”

Tatiana Firsova contributed reporting from Berlin.