For nearly eight decades in postwar Germany, the mainstream political establishment has refused to work with parties it deems extreme. This agreement among centrist leaders is known as a “firewall.” The people behind it say they are trying to prevent potential authoritarians from gaining power as Hitler did. That argument now faces its biggest test in the wake of Sunday’s election in Saxony-Anhalt, a small state of 2.1 million people in the east that broke decisively for the far-right Alternative for Germany party, or AfD.

The result was an example of a fundamental disagreement among Germans about what constitutes a risk to democracy. Voters were asked by pollsters on Sunday if they considered the AfD a danger to democracy, as Germany’s mainstream parties do. About half saw the AfD as a threat, according to exit polling released by a public television network.

They voted almost exclusively for its rivals, from the far left to the center right. The other half of voters said there was no threat. Nearly all backed the AfD.

It was a vivid illustration of the growing divide between Germany’s mainstream parties, which govern from the center-right and the center-left, and populist voters who have grown convinced that politics as usual can no longer solve their urgent problems. To the political mainstream, the far right is a threat to democracy. AfD leaders have made dog-whistle comments about Nazism, including by repeating Nazi slogans and diminishing the significance of the Holocaust, and portrayed minorities as second-class citizens.

To the AfD’s supporters, particularly in the country’s economically struggling eastern states, the AfD has become a last, best shot for making their voices heard — a feeling only reinforced by governing parties’ refusal to work with it. Those voters express anxieties over high levels of immigration from the Middle East over the last decade and a series of deadly attacks by immigrants across Germany in recent years. They complain about high prices for electricity and fuel and dim job prospects, and they still harbor resentments from their treatment by western Germans after the country reunified in 1990.

They say they are less worried that the AfD might infringe on some Germans’ rights, and more concerned that mainstream parties are silencing AfD voters. “I think it’s very good if they get into government because now they can prove that they are not as bad as everyone says, that they are actually not a Nazi party,” Heike Reichelt, a 57-year-old pharmaceutical worker and AfD supporter, said on Sunday evening in Magdeburg, Saxony-Anhalt’s capital. The final vote tally in Saxony-Anhalt showed a dominant AfD performance, but no guarantee it will govern. The party won just under 44 percent of the vote amid record turnout, and 39 seats in an 83-seat Parliament.

Even coming from a relatively small state nestled in the AfD’s eastern power base, the result jolted German politics and shook the concept of a firewall — at a time when the AfD leads national polls. Roger Stöcker, a political scientist at Otto von Guericke University in Magdeburg, called it “a historic turning point.” In the state of Saxony-Anhalt, the AfD could form a government by persuading a small populist party to join it or by poaching renegade members of the center-right Christian Democrats, the party of Chancellor Friedrich Merz. At the national level in Berlin, Mr. Merz is likely to face new questions inside his governing coalition.

Some may push to entertain future partnerships with the AfD. Others could press for more drastic isolation measures, like banning the AfD from federal politics. Across the country, analysts are asking whether the existing strategy can hold.

They note that the public has lost trust in gatekeepers.