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Victory for Mrs Clinton was widely expected but it gives her momentum ahead of the "Super Tuesday" primaries in 11 states this week.
"Tomorrow this campaign goes national," she told cheering supporters.
Mr Sanders has congratulated her but said the campaign was just beginning.
With almost all the votes counted Mrs Clinton leads Mr Sanders by an almost 50-point margin.
Eight out of 10 black voters backed Mrs Clinton, exit polls suggested, a key section of the Democratic electorate.
It is Mrs Clinton's third victory in four contests, after wins in Iowa and Nevada. She lost to Mr Sanders in New Hampshire.
Read more: On the campaign bus with Hillary Clinton
Eight years ago, she lost the South Carolina primary overwhelmingly to then Senator Barack Obama.
It was a different story this time. Soon after polls closed she told supporters: "You sent a message - in America when we stand together, there is no barrier too big to break."
Rivals turn on Trump
On the Republican side, billionaire Donald Trump leads a field that has dwindled to five from 12 a month ago.
He won the Nevada caucus on Wednesday by a wide margin - correspondents say he is beginning to look unstoppable.
In her victory speech, Mrs Clinton aimed a dig at the man tipped to be the Republican presidential candidate.
"Despite what you hear, we don't need to make America great again. America has never stopped being great," she said, referencing Mr Trump's campaign slogan.
Mr Trump's closest challengers in the Republican field, Marco Rubio and Ted Cruz, sought to put him under pressure by releasing several years of their tax returns.
The property magnate says he will not release his until an audit has been completed; his rivals accuse him of holding back the information to hide exaggerations about his wealth.
Mr Sanders, a veteran senator from Vermont, said he was now focussing on the Super Tuesday vote.
"In politics, on a given night, sometimes you win, sometimes you lose. Tonight we lost," he told reporters in Minnesota, one of the states taking part.
"I congratulate Secretary Clinton on her very strong victory. Tuesday, over 800 delegates are at stake, and we intend to win many of them."
But there was some welcome news for Mr Sanders after he was endorsed by Robert Reich, a former official in Bill Clinton's presidential administration.
Key dates to come
1 March - "Super Tuesday" - 15 states or territories decide
18-21 July - Republican convention, nominee picked
25-28 July - Democratic convention, nominee picked
8 November - US presidential elections
In depth: Primary calendar
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