Ms. Wasserman Schultz announced her resignation after a private meeting with advisers and senior aides to Mrs. Clinton at a hotel here a day before the party’s convention was set to begin. She had faced growing calls for her resignation over the weekend.

“In politics, you need to not only know when to draw your sword, but also when to fall on it,” said James Carville, a longtime friend and adviser to the Clintons.

The breach of the Democratic committee’s emails, made public on Friday by WikiLeaks, offered undeniable evidence of what Mr. Sanders’s supporters had complained about for much of the senator’s contentious primary contest with Mrs. Clinton: that the party was effectively an arm of Mrs. Clinton’s campaign. The messages showed members of the committee’s communications team musing about pushing the narrative that the Sanders campaign was inept and trying to raise questions publicly about whether he was an atheist.

Mr. Sanders said the situation was an “outrage” on Sunday before the resignation was announced, and called for Ms. Wasserman Schultz to step down. Afterward, he said it was the right decision.

“The party leadership must also always remain impartial in the presidential nominating process, something which did not occur in the 2016 race,” he said in a statement.

Mrs. Clinton’s campaign aides ignored questions as they quickly left a hotel a few minutes after the resignation was announced. Ms. Brazile emerged soon after the Clinton aides had left and said in a brief interview that Ms. Wasserman Schultz had called her Sunday afternoon and asked her to come to the hotel where the Florida delegation was staying.

Convention organizers had expressed nervousness on Sunday about the specter of Ms. Wasserman Schultz appearing onstage at all during the four-day convention. They were worried that what they intended to be a well-choreographed event, which officials hoped would contrast with the sometimes chaotic Republican National Convention, could be marred by Mr. Sanders’s backers booing and heckling her.