Clinton was gaining roughly half the vote in a three-way Democratic race, with Obama at about 45 percent and former North Carolina Sen. John Edwards placing a distant third.
Romney said Republicans had cast their votes for change - and that he was the man to provide it.
"With a career spent turning around businesses, creating jobs and imposing fiscal discipline, I am ready to get my hands on Washington and turn it inside out," he said in a statement issued while he flew to Florida, site of the Jan. 29 primary.
The Republican caucuses drew relatively little candidate interest. Not so the party's South Carolina primary, the second half of a campaign doubleheader, and a duel between Sen. John McCain and former Arkansas Gov. Mike Huckabee.
Romney was gaining more than half the vote in Nevada, leaving McCain and Texas Rep. Ron Paul in a close race for a distant second place.
Obama had pinned his hopes on an ourpouring of support from the Culinary Workers Union, which endorsed him last week. But it appeared that turnout was lighter than expected at nine caucuses established along the Las Vegas Strip for the union membership.
The Nevada Democratic contest was intense, despite the absence of negative television commercials.
The Clinton campaign said their supporters in the union had been the targets of threats designed to keep them from attending caucuses. Obama's camp said their backers were receiving telephone calls that made repeated reference to "Barack Hussein Obama."
Romney's western victory marked a second straight success for the former Massachusetts governor, coming quickly after a first-place finish in the Michigan primary revived a faltering campaign.
Nevada Republicans said the economy and illegal immigration were their top concerns, according to preliminary results from surveys of voters entering their caucuses. Romney led among voters who cited both issues.
Mormons gave Romney about half his votes. He is hoping to become the first member of his faith to win the White House. Alone among the Republican contenders, Rep. Ron Paul of Texas aired television ads in Nevada.
The first scattered returns showed Romney with more than 50 percent of the vote. Paul, McCain and Huckabee were tightly bunched, far behind the leader.
He also won at least 14 of the 31 Republican National Convention delegates at stake.
Nevada offered more delegates - 31 versus 24 - but far less appeal to the Republican candidates than South Carolina, a primary that has gone to the party's eventual nominee every four years since 1980.
That made it a magnet for former Tennessee Sen. Fred Thompson, who staked his candidacy on a strong showing, as well as for Romney, McCain, the Arizona senator; and Huckabee, the former governor of Arkansas.
McCain, a former Vietnam prisoner of war, appealed to a large population of military veterans in South Carolina, and stressed his determination to rein in federal spending as he worked to avenge a bitter defeat in the 2000 primary.
Huckabee reached out to evangelical Christian voters, hoping to rebound from a string of disappointing showings since his victory in the Jan. 3 Iowa caucuses.
Romney campaigned on a pledge to help restore the state's economy, much as he did in winning Michigan.
In South Carolina, the economy and immigration were cited as top issues, and preliminary survey data indicated a strong turnout by evangelical voters.
Survey data in both states were from polls conducted for The Associated Press and the television networks by Edison Media Research and Mitofsky International.
South Carolina primary voters coped with equipment difficulty and bad weather.
Election officials in the area around Myrtle Beach brought out paper ballots after some electronic voting machines failed to work properly.
Snow fell in the northern part of the state, which has little snow removal equipment.
Alone among the major Republican contenders, former New York Mayor Rudy Giuliani skipped the day's events. He camped out in Florida, the first of the big states to vote, with a winner-take-all primary.
Obama and Clinton both ran all-out in Nevada, even though only 25 delegates were at stake.
Preliminary results from interviews with voters entering their caucuses showed woman voting heavily for the former first lady. She and Obama split the male vote.
Obama's backing from the influential Culinary Workers Union led to an unsuccessful lawsuit by some of Clinton's supporters who hoped to ban specially arranged caucuses along the Las Vegas Strip that could draw thousands of unionized casino and hotel workers.
Both campaigned to the last moment among casino and hotel workers. Edwards spoke to a union audience in Missouri, then was heading to South Carolina, where Democrats hold their own primary in one week.
Obama, hoping to become the first black president, spent nearly $1 million on television commercials. Clinton, campaigning to become the country's first woman chief executive, ran nearly $700,000 worth of commercials, and a union group backed her with nearly $100,000 spent on an independent ad campaign.
Former President Clinton was a constant presence, as well, in a state he carried twice on his own in 1992 and 1996.
Remarkably, neither Obama nor Clinton has aired a television commercial criticizing the other, and both of the rivals stepped back earlier in the week from a controversy over race. But that didn't prevent almost constant sniping between the two camps, each pointing out alleged inconsistencies in the other's record.
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Associated Press writers Seanna Adcox and Libby Quaid, both in Columbia, S.C., Bruce Smith in Mount Pleasant, S.C., and Glen Johnson in Las Vegas contributed to this report.
