MANCHESTER, N.H. — Donald Trump was projected as the winner of the New Hampshire Republican presidential primary Tuesday, easily outdistancing his rivals, claiming his first victory and firmly establishing himself as the candidate to beat.

The billionaire businessman ran away with the contest, winning across all demographics, including committed Republicans and independents, conservatives and moderates, and regular voters as well as first-timers.

He doubled the tally of the second-place finisher, Ohio Gov. John Kasich. Behind Mr. Kasich were Sen. Ted Cruz of Texas, former Florida Gov. Jeb Bush and Sen. Marco Rubio of Florida.

The results did little to settle the question of who will emerge as the anti-Trump Republican establishment pick, though Mr. Kasich’s showing gives him an edge moving forward and Mr. Rubio is likely to be dented.

New Jersey Gov. Chris Christie, who was headed for a sixth-place finish, may be a casualty of the vote.

He said he was going home to ponder the results, after he failed to turn a well-received debate performance this weekend into votes here.

Mr. Trump, meanwhile, showed he could draw from registered Republicans and independents, and from regular voters and first-time voters.

As the race turns southward and westward, Mr. Trump is well-positioned to capitalize, leading in polling in many of the next big states to vote. At his victory party, he predicted that New Hampshire was just the beginning of his run of victories.

“It’s going to be for many weeks, OK?” he said, thanking New Hampshire voters for sending him out on top. “You started it. Remember. You started it.”

His supporters didn’t recoil from his brash style but embraced it, saying it was a big reason they voted for him.

“He’s different. He says things nobody else does,” said Peter Dite, an airport dispatcher and immigrant from the Czech Republic who backed the maverick businessman — even as his wife, Robin, voted for Sen. Bernard Sanders in the Democratic primary.

The couple said Mr. Sanders and Mr. Trump were tapping into similar angry strains in the American electorate.

Mr. Kasich, meanwhile, said his second-place showing was the counterpoint to Mr. Trump and the rest of the field.

“Tonight, the light overcame the darkness of negative campaigning. You made it happen,” the governor said.

With about 73 percent of precincts reporting, Mr. Trump had 35 percent of the vote, followed by Mr. Kasich with 16 percent. Mr. Cruz had 12 percent, while Mr. Bush had 11 percent and Mr. Rubio 10 percent. Mr. Christie had slightly less than 8 percent, for a sixth-place showing.

That was particularly disappointing for the New Jersey governor, whose less-than-stellar showings in Iowa and New Hampshire may have fatally wounded him.

He said Tuesday night that he would go back to New Jersey to take stock of his numbers and see where he fits in the field. His sixth-place finish means he likely would not appear on the stage for the next debate anyway.

Mr. Christie had hoped voters would would reward him for the verbal spanking he delivered to Mr. Rubio in Saturday night’s debate, when the first-term senator stumbled over a challenge from the New Jersey governor.

Mr. Christie said Mr. Rubio came across as robotic, repeating the same talking points. Some New Hampshire voters said they were swayed to Mr. Christie by the debate.

“I went into the debate, I said, ‘I am either going Rubio, Bush or Christie,’ and I really wanted to go Rubio. … But it was bad,” said Steve Kelliher, 56, voting in Bedford.

He ended up pulling the lever for Mr. Christie, saying the governor has proved he could work with a Democratic legislature in New Jersey, and that his blast against Mr. Rubio proved he was the fighter the party needs.

“That is the only reason [I like Mr. Christie]. I like his approach and I can project him going up against Hillary and doing a really good job in a debate and maybe swaying people,” Mr. Kelliher said.

But exit polling suggested that while Mr. Rubio dropped, Mr. Christie wasn’t picking up those voters.

At his post-vote speech, Mr. Rubio took blame for his debate performance.

“That will never happen again,” he assured his supporters.

“Not all the days are going to be great days. We’re not always going to get things the way we want,” Mr. Rubio said. “But in the end, I am confident that not only will this campaign be successful, but America will be successful as well.”

For his part Mr. Cruz, the winner of last week’s Iowa caucuses, said he was happy with his showing here in New Hampshire, saying he defied those who doubted him.

“Together we have done what the pundits and the media elites said couldn’t be done and what the Washington establishment desperately hoped would not be done,” he said. “The real winner is the conservative grass roots who propelled us to an outright victory in Iowa and a far stronger finish in New Hampshire than anyone predicted. And now we go on to South Carolina.”

Mr. Bush insisted his showing was good enough to keep him in the race, too.

“This campaign’s not dead,” said the former governor.

Mr. Trump’s surge, meanwhile, has become the biggest story of the 2016 election, as he’s confounded pundits and strategists with his ability to survive gaffes that would have doomed other candidates.

After a disappointing second-place showing in Iowa, which Mr. Cruz won, Mr. Trump said he probably should have done more to build a ground organization there.

On Tuesday, Mr. Trump said he learned quickly.

“You know, we learned a lot about ground games in one week, I have to tell you,” he said before reiterating his vows to build a wall along the U.S.-Mexico border, wipe out the Islamic State terrorist group and be the “greatest jobs president that God ever created.”

“We are going to start winning again, and we are going to start winning so much, you are going to be so happy. We are going to make America so great again — maybe greater than ever before,” he said.

His polarization was evident even within families, including Roger Cross and his wife. As they emerged from a polling place in Concord, Mr. Cross, 68, said he voted for Mr. Trump, sparking a playfully indignant response from his wife.

“Oh my God, you are admitting you are an idiot,” she said as she fled to their car.

He shot back, “Will you be quiet? I am talking!”

“I am going to the car before I hit you!” she said.

He countered, “Go to the car! Go to the car, get away from me, God.”

As for why he supported Mr. Trump, he said, “He is a massively wealthy businessman that wants to go in and shake this up.”

As for his wife, he said, he suspected she voted for Mr. Sanders.

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