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Supporters of Senator Bernie Sanders are pushing back against the narrative that he played second fiddle to Hillary Rodham Clinton in Tuesday night’s Democratic debate.

The self-proclaimed Democratic socialist from Vermont has been heralded for being magnanimous in his handling of a question about Mrs. Clinton’s use of a private email server when she was secretary of state, and he was widely praised for offering principled responses about criminal justice and inequality.

But pundits and commentators largely gave the victory to Mrs. Clinton, suggesting that her crisp performance could assuage concerns about her candidacy and quell calls for Vice President Joseph R. Biden Jr. to enter the race.

Are they right? Not according to Mr. Sanders’s fans, who point to an array of online surveys and metrics that indicate he got the biggest boost from his performance.

“Bernie won the debate, yet the corporatists say otherwise,” Claudia Chiuaru wrote on Facebook, next to an article that suggested Mrs. Clinton had the better night. “This is insane!”

Mr. Sanders is known for having an eager army of Facebook backers who promote his cause. Several unscientific online polls did show that Mr. Sanders was the overwhelming winner of the debate, and his campaign offered its own evidence to support that.

“On social media, where Sanders’s grass-roots revolution began, there were more Google searches for Sanders than for any other candidate,” said Michael Briggs, a spokesman for the Sanders campaign. “He was the most retweeted candidate of the night, according to Twitter. He gained more followers on Twitter than any other candidate and Twitter said people talked about Sanders more than any other candidate online.”

Mr. Sanders’s camp also pointed to strong fund-raising totals after the debate as an indication that his supporters were energized by his performance.

Data from CrowdTangle, a social media tracker, found that Mr. Sanders did have a good night online. He gained the most Facebook followers on Tuesday night (35,000) and the “Sanders for President” thread on Reddit was the most active.

Some critics of Mr. Sanders said that he appeared out of his comfort zone when discussing foreign policy and while defending his record on guns. But according to Facebook, which co-sponsored the debate, Mr. Sanders managed to achieve the social media “moment” of the evening when he tried to put Mrs. Clinton’s email issues to rest.

“What the secretary said is right,” Mr. Sanders said. “And that is the American people are tired of hearing about your damn emails.”