“I am so so thrilled and so grateful to all of my supporters out there,” she said at a victory rally. “Some may have doubted us but we never doubted each other.”
With 81% of precincts reporting, Mrs. Clinton had a 52.1% lead compared with 47.8% for the Vermont senator.
The victory was a major relief for the Clinton campaign and allows the former secretary of state to move on to contests in South Carolina next Saturday and the states that follow with fresh momentum after being walloped in New Hampshire and winning by only the slimmest of margins in Iowa.
Mrs. Clinton had long been considered the favorite in Nevada. But Mr. Sanders’ 22-point victory margin in New Hampshire gave him significant momentum. An infusion of donations also allowed him to put staff on the ground and television ads on the air.
“I am also proud of the fact that we have brought many working people and young people into the political process and believe that we have the wind at our back as we head toward Super Tuesday. I want to thank the people of Nevada for their support that they have given us and the boost that their support will give us as we go forward,” Sanders said.
In the hours before the caucuses started, Mrs. Clinton visited the Harrah’s employee cafeteria in Las Vegas late Saturday morning about 20 minutes after Mr. Sanders passed through the same room. “Hillary! Hillary! Hillary!” the room cheered as she entered.
Her husband, former President Bill Clinton, was nearby at the Caesars Palace hotel on the Las Vegas strip. “We’re doing everything we can,” he said. “We got a good response.”
Las Vegas hotel workers, many in uniform, swarmed a Democratic caucus site in a Caesars Palace ballroom, one of six at-large caucus locations in the city aimed at making it easier for shift workers to take part in the contest.
Waiting to get into the ballroom, Michael Jaumotte, a food service worker at Caesars, said he was backing Mr. Sanders. “I like that he’s not taking donations from big corporations,” he said. A group of housekeepers from the Bellagio Hotel arrived together wearing their brown uniforms and “Hillary” stickers.
Walking into his polling precinct at a middle school in Henderson, Nev., Senate Democratic leader Harry Reid said he would vote uncommitted, to stay neutral in the race. “Everyone knows the caucus setup here in Nevada are my doing,” said Mr. Reid, who was instrumental in moving the Nevada caucuses to the forefront of the nominating process.
“I want to make sure that Clinton and Sanders know that I’ve been fair. If I got involved on one side or another, it would be easy for them to say, you know he’s trying to affect the caucuses unfairly,” he said, adding that he would offer an endorsement after the caucuses are over.
Both Mrs. Clinton and Mr. Sanders made a point of spending at least part of their time in the days before the caucuses in the mountains and deserts of Nevada, away from the glam and glitter of Las Vegas.
Mr. Sanders spent Friday stumping in Elko and Reno before returning to Las Vegas for a large rally. Mrs. Clinton made stops in both Reno and Elko earlier in the week.
Consequently, the road to victory in Saturday’s Democratic caucuses runs through places like Elko, a small mining community that is the closest thing to a major city in northeast Nevada, and Pahrump, a desert town of about 36,000 that boasts a handful of casinos, a couple of brothels—and Clinton and Sanders campaign field offices.
And both campaigns worked these sparsely populated communities, phoning supporters daily and driving to remote houses on dirt roads that aren’t on the maps to knock on doors and see if the people who answer might need a lift to caucus sites.
Mr. Sanders brought his pitch to Elko on Friday, a picturesque mountain town of 20,000 people that lies more than 300 miles north of Las Vegas in a conservative part of the state. He was following Mrs. Clinton, who flew in earlier in the week. Republican candidates, who hold their caucuses Tuesday, were expected to start arriving in northern Nevada over the next few days.
But the Vermont senator finished up his Nevada campaign late Friday in Henderson, Nev., the second largest city in the state. He didn’t predict victory in his speech, but instead forecast Saturday as a key date in his hopes for the presidency.
“It could well be that 10, 20, 30 years from now people will look back on what happens in Nevada and say this was the beginning of the political revolution,” Mr. Sanders said.
Though the overwhelming majority of the state’s people live in and around Las Vegas, campaigns ignore the rural areas at their own peril. In the 2008 Democratic race between Mrs. Clinton and Barack Obama, she won the state but Mr. Obama collected more delegates in part because of a stronger organization and presence in the rural areas.
That was a mistake her campaign didn’t plan to repeat. She has offices in Elko, Pahrump and Carson City, and organizers across the state. Mr. Sanders has 12 field offices, though his campaign started organizing in Nevada months after Mrs. Clinton arrived.
The Clinton campaign says its staffers logged 1,250 miles across rural Nevada last summer, sending senior campaign leaders into communities like Ely, Winnemucca, Hawthorne and Tonopah for a listening tour that helped shape the campaign’s rural strategy.
Bill Clinton also visited Pahrump this month and spoke to a few hundred people at a local elementary school.
Gabe Alvare, a 24-year-old Clinton volunteer visiting from New York City, spent hours knocking on doors of known supporters on Wednesday in Pahrump, making sure they planned to show up for the caucus. He had stopped at many of the same houses a few days earlier. He offered Clinton yard signs, but there were no takers. Nye County is predominantly Republican, and some of the people seemed reluctant to post a yard sign that might antagonize neighbors.
At the wheel of the GMC pickup truck in which Mr. Alvare traveled was Clinton supporter Frank Dix, 73, who lives in Pahrump and feels a special loyalty to the Clinton family.
“I’m partial to strong women,” Mr. Dix said. “Hillary dealt with Bill. That’s what my wife says about me. She says, ‘I had to be strong, because otherwise Frank would run all over me.’ ”
Navigating can be a challenge. The recession hit Pahrump hard, and construction on some roads seemed to stop before completion.
“We go where the houses are, not where the roads are,” Mr. Dix said.
There is no surefire way to get people to turn up at the caucus sites. One person mentioned that softball practice posed a potential conflict. A woman said her husband was planning to go, but he hasn’t been feeling good lately.
There have been some reassuring moments for Messrs. Alvare and Dix. Helaine Jennings, a 69-year-old retiree, answered her door and said she was all-in for Mrs. Clinton. When Mr. Alvare asked if she needed a ride, she declined. “We have two legs,” she said. “It’s very close.”
At a spartan office nearby, the Sanders campaign was working to get out the vote. Rick Bandell, a 75-year-old retiree, said he was thrilled to call one home and persuade an entire family to show up in support of Mr. Sanders.
Republicans outnumber Democrats in Nye County, which includes Pahrump, by about 3 to 2, and the GOP complexion of the town is evident. At Mom’s Diner, next to the Clinton campaign office, 10 Pahrump supporters of GOP candidate Ted Cruz were meeting for lunch and plotting strategy.
What was also clear was that people of Pahrump were dialed in to the race.
At the Lakeside Casino and RV Park, 63-year-old Leanette Robertson paused from her video-poker game to discuss the Sanders-Clinton contest. She favors Mrs. Clinton but wasn’t sure she would be able to go to the caucuses because she had family visiting.
“She’s got a broad spectrum of issues she’s supporting,” Ms. Robertson said of Mrs. Clinton. “It’s not just the money issue. Bernie seems to be focused on one issue: the 1% of the 1%.”
—Reid J. Epstein contributed to this article.