But the rough-and-tumble campaign claimed another victim when former Florida governor Jeb Bush - brother and son to two US presidents - dropped out of the race after another poor showing. In South Carolina, the 69-year-old Trump captured about a third of the votes, with all of the precincts reporting.
His supporters erupted in a roar when CNN called the contest in his favour - his second win of the nominations race after New Hampshire and an important test of the strength of his bid to succeed President Barack Obama. "There is nothing easy about running for president, I can tell you," Trump told his victory party in Spartanburg. "It's tough, it's nasty. It's mean. It's vicious. It's beautiful."
"When you win, it's beautiful."
After several nail-biting hours, final results showed Florida Senator Marco Rubio in second place in the Republican contest with 22.5 percent of the vote, narrowly ahead of Texas Senator Ted Cruz, who had 22.3 percent.
"After tonight, this has become a three-person race and we will win the nomination," said Rubio, like Cruz a Cuban-American first-time senator.
In Nevada, Clinton claimed a major win in the Democratic race. Final results gave her 52.7 percent of the vote against 47.2 percent for Sanders.
"This is your campaign, and it is a campaign to break down every barrier that holds you back," Clinton said in her victory speech at her Caesars Palace headquarters on the Las Vegas Strip.
In South Carolina, Trump - a onetime reality TV star who has upended the political landscape with his tough talk on everything from Muslims to Mexico to waterboarding - showed he could compete for the long haul.
"We just think we want to take a risk with Trump. We think he's had success with everything he's touched," Lynn Derrick, a first-time primary voter, told AFP in the state capital Columbia.
Trump and Cruz, who won the Iowa caucuses, duked it out in the week leading up to Saturday's primary, with the campaign growing increasingly nasty. Trump even had a spat with Pope Francis, who suggested the tycoon was "not a Christian" for wanting to build a wall on the border with Mexico. Trump also called for a boycott of Apple over its refusal to unlock the phone of a suspect in the San Bernardino attacks.
If Trump wins the Republican nomination and is elected president in the November 8 election, he would be the first US president in history to have no government experience.
Saturday's results were bad news for Bush, Ohio Governor John Kasich and retired neurosurgeon Ben Carson, who were relegated to the second tier of candidates.
Bush did not wait long to fold.
"In this campaign, I have stood my ground, refusing to bend to the political winds," he said.
In the desert state of Nevada, the 68-year-old Clinton scored a major win, but Sanders proved he was in it for the long haul.
"Some may have doubted us, but we never doubted each other," Clinton said in a fiery left-leaning speech clearly aimed at the minority voters who make up roughly half of the state's population and who are key in the US South.
Clinton claims that Sanders is offering impractical, fantasy ideas that he cannot deliver as president, and on Saturday took aim at corporate America - usually a message sounded by Sanders.
"If you cheat your employees, exploit consumers, pollute our environment, or rip off taxpayers, we're going to hold you accountable," Clinton said.
"But if you do the right things, if you invest in your workers, contribute to your communities, help build a better America - we're going to stand with you."
Clinton, who won by a hair in Iowa but was crushed by Sanders in New Hampshire, had counted on a major Hispanic voter turnout, especially among Las Vegas hotel and casino employees. CNN entrance polls showed that she handily won the black vote, but the Latino vote was more evenly split - evidence that the Sanders campaign may have more stamina than first expected.
Clinton did well with women, but was again stomped by Sanders with young voters, the polls showed.
Sanders congratulated Clinton, and said he was proud of having significantly narrowed the gap.
"We have the wind at our back as we head toward Super Tuesday," the 74-year-old Sanders said, looking ahead to the showdown, when about a fifth of the Democratic nominating delegates are in play.