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 Sen. Bernie Sanders of Vermont as their fight turns to the 11 states where Democrats vote 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 Clinton here. They supported her over Sanders by a 5-to-1 ratio, while he won the bulk of white voters, according to early exit polling.

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," Clinton said, a line she often used in South Carolina, where Obama defeated her by 29 points in 2008.

This time it was Clinton who emerges from the first southern primary with a clearer path to the nomination. With early exit polls showing 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 with the 532,000 ballots cast in the Clinton-Obama primary race here in 2008, and well below the record-setting 743,000 votes cast in South Carolina Republican primary last Saturday, which Donald Trump won.

The rout was both political and psychologically meaningful for Clinton after she barely defeated 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.

Beyond delivering momentum, the win also extends Clinton's lead over Sanders in delegates needed to clinch the Democratic nomination; the exact allocation of South Carolina's 53 delegates will be known once all the votes are counted.

The results also helped Clinton extinguish any doubts among Democrats about her ability to appeal to black voters, with black women, in particular, enthusiastically backing her candidacy both in South Carolina and in the Nevada caucuses last week.

That reservoir of support will serve as the biggest roadblock to Sanders' chances for a surge in the weeks ahead. Clinton advisers believe she will trounce Sanders in Alabama, Arkansas, Georgia, Texas, Tennessee and Virginia, which have contests Tuesday, and, in doing so, move even further ahead of him to capture the 2,383 delegates needed to clinch the Democratic nomination.

Under party rules, most delegates are awarded proportionally to Clinton and Sanders based on their shares of the vote in congressional districts. The most Democratic-leaning districts are accorded the most delegates; in many places these are majority black and Hispanic districts, and Clinton is far more popular with those voters than is Sanders.

"Support from African-Americans is going to be the key for Secretary Clinton across the South, and South Carolina is a good indication of that," said former Gov. Richard Riley, who was education secretary under President Bill Clinton and is a supporter of Clinton. "She has supported President Obama in a very serious way, and worked all out for him as secretary of state, and that matters a lot to many Democratic voters."

As if already looking past South Carolina, which had for months heavily favored Clinton, Sanders addressed big crowds at two rallies in Texas on Saturday. His aides believe Clinton remains vulnerable and that Sanders will pick up delegates on March, 1, known as Super Tuesday, when states including Oklahoma, Massachusetts, Minnesota, Colorado, and his home state of Vermont hold nominating contests.

One glaring problem for Clinton was reaffirmed in South Carolina: Her shaky support among white voters. According to exits polls by Edison Research, Sanders won a majority of them, beating her among white men and white college-educated Democrats. While Clinton lost among white voters in the last two states, New Hampshire and Nevada, South Carolina allies like former Gov. Jim Hodges had said they were confident she would win them here, noting that the state had relatively fewer young white liberals than in the other states.

But in interviews last week, several white Democrats were sharply critical of Clinton and said they did not trust her, chiefly because of her use of a private email account and the U.S. deaths in Benghazi, Libya, when she was secretary of state. Some Clinton allies are deeply concerned that she could struggle with white voters, particularly white men, in the general election, and Republicans like Trump have made clear that they intend to crush her among white Americans.

The senator had tried to make inroads among African-American voters, including highlighting a grainy black-and-white photo published in The Chicago Tribune that showed a young Sanders being arrested in Chicago in 1963 while protesting segregation.

In a radio ad that aired here, the director Spike Lee urged the state's black voters to "Do the right thing!" and elect Sanders. "Bernie takes no money from corporations. Nada. Which means he is not on the take," the director said, a jab at Clinton's paid speeches and perceived ties to Wall Street.

The Democrats have each emphasized a commitment to overhauling the criminal justice system. This week Clinton campaigned with the mothers of Trayvon Martin, Eric Garner and other black men who died because of gun violence or in clashes with the police.

On Tuesday, Sanders told CNN, "There is something very wrong when African-Americans in South Carolina and around the country get nervous walking down the street or going into their car and being stopped by a police officer."

But in the end, Sanders' populist economic message proved no match for Clinton's deep ties in the state.

From her first trip here last spring, Clinton has emphasized her friendship with Obama and her commitment to protecting his legacy. And, as the race tightened, she tried to portray Sanders, who was an independent until April, as a critic of Obama who wanted to dismantle the Affordable Care Act.

The Sanders campaign called that unfair, noting that his plan for universal health care would improve upon the gains made under the law, but it achieved Clinton's goal of painting the senator as more interested in lofty ideals than in safeguarding the president's signature achievement.

Both she and Bill Clinton heaped praise on Obama during their recent stops in the state, in jarring contrast to the South Carolina primary eight years ago. In that race Bill Clinton infuriated African-Americans here by lashing out at the Obama campaign and comparing Obama's success to the victories here by the Rev. Jesse L. Jackson in the 1984 and 1988 primaries. Many Democrats, especially black voters, viewed those comparisons as diminishing to Obama.

The Clintons had worked hard to repair any damage done in 2008, and Saturday's win — and the strong support from black voters — was particularly satisfying to Bill Clinton, friends said. While he felt his remarks then were misinterpreted, Clinton valued his decades-old bond with southern black voters and had been distressed when they deserted Hillary Clinton in such large numbers eight years ago.

Going into the South Carolina primary, Clinton had 52 pledged delegates to 51 for Sanders. Pledged delegates are the kind awarded proportionally in state primaries and caucuses.

Clinton also had backing from 453 superdelegates — party officials whose support counts toward the 2,383 to win the nomination — while Sanders has support from 20. Superdelegates can switch candidates at any time, and their support doesn't become official until the Democratic convention in July.

About 880 Democratic delegates are at stake Tuesday, the largest number on any single day during the primary season.

On Sunday, Clinton planned to campaign in Memphis and Nashville, Tennessee, and Pine Bluff, Arkansas, a poverty-stricken city in Clinton's home state that is more than 75 percent African-American.

Even before the results of the South Carolina primary had been tallied, Clinton held a rally Saturday in Alabama, which will vote Tuesday. "Fired up! Ready to go!" the crowd chanted, using a popular line from Obama's 2008 campaign.

"I know there are a lot of people in this state who want to continue the progress we've made, who want to keep moving forward," an upbeat Clinton told the crowd.

Before the rally, she stopped at the Yo' Mama's chicken and waffles restaurant and the Urban Standard coffee shop, where she declared, "Down the hatch!" as she downed a "straight up" shot of espresso.

"I like your style," the barista said.