COLUMBIA, S.C. — Drawing overwhelming support from the African-American voters who deserted her here eight years ago, Hillary Clinton won her first resounding victory of the 2016 campaign in South Carolina on Saturday, delivering a blow to Senator Bernie Sanders of Vermont as their fight turns to the 11 states where Democrats vote on Tuesday.

After supporting Barack Obama in 2008, African-American voters, who will be the dominant force in the coming Southern primaries, turned out in droves for Mrs. Clinton here. They supported her over Mr. Sanders by a 5-to-1 ratio, while he won the bulk of white voters, according to early exit polling.

Mrs. Clinton assiduously cultivated the support of black voters, not least by showing her devotion to President Obama and by promising to build on his legacy. She capped off months of campaigning here with stops on Friday at a popular soul food restaurant and bakery in Charleston and a rally at a historically black college in Orangeburg, alongside black surrogates including the TV personality Star Jones and the state’s longtime representative, James E. Clyburn.

“I don’t think President Obama gets the credit he deserves for digging us out of the ditch Republicans put us in,” Mrs. Clinton said, a line she often used in South Carolina, where Mr. Obama defeated her by 29 points in 2008.

This time it was Mrs. Clinton who emerged from the first southern primary with a clearer path to the nomination. With early exit polls showing Mrs. Clinton winning by more than 35 percentage points, The Associated Press called the primary for her shortly after polls closed at 7 p.m. State officials projected that turnout was modest compared to the 532,000 ballots cast in the Clinton-Obama primary race here in 2008, and well below the record 743,000 votes cast in South Carolina Republican primary last Saturday, which Donald J. Trump won.

The rout was both political and psychologically meaningful for Mrs. Clinton after she barely defeated Mr. Sanders in Iowa, lost to him by 21 percentage points in New Hampshire, and eked out a 5.5 percentage point victory last week in Nevada.