It was the 10-word interjection that upended the trajectory of the night, if not the 2020 campaign so far.

“I would like to speak on the issue of race,” Ms. Harris declared.

The room soon went silent.

Ms. Harris turned to address Mr. Biden, directly and personally, marrying her own identity as an African-American woman with a pointed critique of not just his recent rhetoric about working with segregationists but what they worked on together. “You also worked with them to oppose busing,” she said. “And you know, there was a little girl in California who was a part of the second class to integrate her public schools, and she was bused to school every day. And that little girl was me.”

Mr. Biden protested. “A mischaracterization of my position,” he said.

She pressed on, framing her follow-ups as the prosecutor she once was. “Do you agree today that you were wrong to oppose busing in America then? Do you agree?”

In that raw and intense moment, Ms. Harris flashed the political potential that many Democrats believed she held yet had heretofore not been realized. And she did so while boldly taking on the leading Democratic in the race, Mr. Biden, whom many in the field have shied from confronting because of his residual popularity from eight years as Mr. Obama’s No. 2.