The victory in South Carolina will give Clinton momentum as the contest heads toward Super Tuesday (Wednesday NZT), where she and Sanders will compete in 11 states.
A slightly hoarse Clinton came out to cheers of "Hillary! Hillary!" in a room full of jubilant supporters in Columbia, the state capital.
"Today, you sent a message: In America, when we stand together, there is no barrier too big to break," Clinton said. Acknowledging that South Carolina was the end of the one-state-at-a-time primary contests, she exclaimed: "Tomorrow, this campaign goes national!"
Exit polls reported by ABC News showed that Clinton's advantage with black voters was decisive.
Sanders vowed to fight on: "Let me be clear on one thing tonight. This campaign is just beginning."
He later said: "[On Wednesday], over 800 delegates are at stake, and we intend to win many, many of them."
Sanders has won only one of the four initial contests, where candidates typically focus the greatest attention and resources. But he has captured young voters in astonishing numbers and is raising more money than Clinton, ensuring that he will remain in the race - and remain a threat.
A thousand kilometres from South Carolina in Austin, Texas, more than 10,000 adoring Sanders supporters showed up and cheered his every sentence at an outdoor rally. He had plenty to say about elections - just not the one taking place yesterday.
Sanders could not make headway against Clinton's long ties with and enduring loyalty from black voters. He had hoped to do well enough to claim he had dented Clinton's "firewall" of Southern states, which Clinton allies have claimed would put an end to Sanders' early momentum from liberal, majority-white states.
"We've made a lot of progress in the last eight years, and Hillary is the best person out there to continue the progress," said Al Tucker, a 67-year-old African American in Columbia.
"You look at South Carolina, and we're at the bottom in anything you can think of: education, poverty. I think Hillary would be good because she's going to look out for us."
Clinton reiterated her criticism of the rhetoric of Republican front-runner Donald Trump, particularly toward Muslim Americans. During a stop in Birmingham, Alabama, the former Secretary of State said: "When you run for President, it's not just Americans who pay attention. And when you are President the entire world listens to every word you say. Markets rise and fall. You do have to be careful about what you say and how you say it."
Clinton relied on decades of relationships between her family and black leaders in South Carolina. Supporters at her rallies last week were largely older African Americans.
Many of those same supporters had spurned Clinton in 2008 to support then rival Barack Obama.
The Clintons were accused of using racially tinged rhetoric to disparage Obama's candidacy eight years ago. But both Hillary and former President Bill Clinton were redeemed with stalwart black support this time.
Among those supporters was Bernice Scott, 71, who was up early, heading to the small towns around Richland County, where she has lived for nearly 50 years, to help get out the vote for Clinton as part of a network of grass-roots activists known as the "Reckoning Crew".
