Though Mr. Biden had led in every poll of South Carolina, Mr. Sanders, after winning in a landslide in Nevada, decided to try to deliver a finishing blow against Mr. Biden. Mr. Sanders increased his television advertising in the state and intensified his campaign schedule, with the goal of denying Mr. Biden the chance to reignite his candidacy and perhaps wrapping up the nomination fight by the middle of March.

Addressing supporters in Virginia, Mr. Sanders, 78, acknowledged Mr. Biden’s success in South Carolina and advised his audience to prepare for the ups and downs of a long campaign. “That will not be the only defeat,” Mr. Sanders said of South Carolina. “There are a lot of states in this country, nobody wins them all.”

But ticking off his victories so far, Mr. Sanders also pointed in a confident tone toward Tuesday’s primaries as the next frontier.

Ms. Warren, at a rally in Houston, also looked ahead to those contests. “I’ll be the first to say that the first four contests haven’t gone exactly as I’d hoped,” she told supporters. “But Super Tuesday is three days away and we’re looking forward to gaining as many delegates to the convention as we can.”

Having carried South Carolina as a kind of favorite-son candidate, Mr. Biden is counting on that result to ripple throughout the region and help him recover some of the support from black voters elsewhere that he lost in recent months, largely to Mr. Bloomberg. He needs voters to shift back in his direction quickly if he is to edge ahead of Mr. Sanders in enough states to deliver a strong showing on Super Tuesday.

But absent an overwhelming wave of new support for Mr. Biden, the best-case scenario for his campaign may still be a daunting one: a monthslong battle against a tireless opponent with superior financial and organizational resources at his disposal, and a formidable well of support from the Democratic Party’s left wing.

Mr. Sanders has had a weekslong head start in a number of key Super Tuesday states where early voting has long been underway, including California, which on its own could give Mr. Sanders a sizable lead in the national delegate count.