HENDERSONVILLE, N.C. — In his campaign headquarters the morning after his electoral victory, Chuck Edwards showed no interest in dissecting one of the biggest political upsets so far in this year’s Republican primary season.

Mr. Edwards, 61, a three-term state senator and business owner, thwarted Representative Madison Cawthorn’s turbulent re-election bid in North Carolina, beating him in Tuesday’s primary in a rare defeat of a Trump-backed Republican incumbent.

“I’m excited for the opportunity to unify the Republican Party, put the primary behind us and focus our attention towards the real issues,” Mr. Edwards said on Wednesday, seated at a sleek mahogany conference table at his campaign office in downtown Hendersonville.

What went unspoken was that many voters saw him as the establishment candidate who benefited from the boost of old-guard Republicans at home and in Washington. Mr. Cawthorn, 26, had alienated two powerful Republicans with a litany of political and personal errors and scandals: Representative Kevin McCarthy of California, the House minority leader, and Senator Thom Tillis of North Carolina.

A political group supporting Mr. Tillis, who endorsed Mr. Edwards, poured money into an ad campaign that painted Mr. Cawthorn as a fame-seeking liar. Other top North Carolina Republicans, including the state’s House speaker and State Senate leaders, also came to Mr. Edwards’s side.

In most Republican primaries across the country, old-fashioned establishment candidates in the Romney and Bush mold have been on the run amid intraparty battles and challenges from a far-right wing energized by former President Donald J. Trump. But as Mr. Edwards’s low-key postelection demeanor suggested, the state’s establishment seemed unwilling to claim victory when the word establishment itself has become an insult in Republican politics.

Asked about those who saw him as the establishment figure in the race, Mr. Edwards said people had different definitions of the label.

“It is true I have established conservative principles and a track record of getting things done,” he said. “I’ve established that I can cut taxes. I’ve established that I can balance budgets. I’ve established that I can pass bills to outlaw sanctuary cities.”

Of course, Mr. Cawthorn’s defeat was not solely the work of the establishment. He appeared to struggle with two key demographics: unaffiliated voters, who make up more than 40 percent of his district, and those in Henderson County, which includes his hometown of Hendersonville and helped carry him to victory in the last Republican primary.

A barrage of bad press and personal and political mistakes helped turn voters against Mr. Cawthorn. Many said in interviews that they still supported Mr. Trump, but that they viewed Mr. Cawthorn as irresponsible, immature and unbecoming of public office. Mr. McCarthy, for his part, told reporters in March that he had spoken to Mr. Cawthorn after the freshman congressman had implied members of his own party invited him to orgies and to use cocaine with them.

Mr. Cawthorn had been accused of engaging in insider trading, was pulled over for speeding, charged with driving with a revoked license and had been stopped for trying to bring a gun through airport security a second time. Photos and videos of him partying and emulating sexual antics circulated. Most damaging were reports that he frequently missed votes and had abandoned constituency offices.

Mr. Edwards jumped into the race after Mr. Cawthorn announced last year that he would run in a new district near Charlotte. Mr. Cawthorn ended up changing his mind and returning to his old district after the new district was redrawn and tilted Democratic.

Mr. Edwards, who owns several McDonald’s franchises and has served in the state legislature since 2016, has built a staunch conservative brand to the right of what used to be considered the traditional establishment Republican. He has pushed measures to overhaul tax laws, enact a constitutional amendment for voter identification and require county sheriffs to work with immigration enforcement agencies.

In his campaign office Wednesday, days after a racist mass shooting at a Buffalo supermarket, Mr. Edwards said he did not believe limiting gun ownership for law-abiding citizens would solve societal problems. And he declined to condemn the racist conspiracy theory that police say motivated the Buffalo gunman — and that has been echoed by members of his own party. That unfounded theory essentially holds that elites are using immigration and falling birthrates to replace white people and destroy white culture.

“I don’t focus on rhetoric, I focus on results,” Mr. Edwards said. “I condemn open borders and the fact that we’re losing the sanctity of our nation by not enforcing immigration laws at the border.”

On Tuesday night, at a shuttered mechanics shop in Hendersonville that was converted into an event space for Mr. Cawthorn’s election party, the mood had started upbeat as people mingled over snow cones and games of cornhole. Foldout tables were decked with small candy bowls and flower-vase centerpieces. There were party favors for guests: cupcakes, star-shaped sunglasses and beaded necklaces in red, white and blue. A face painter waited for children to stop by her stand.

But as night fell, the crowd grew larger and antsier. People wearily chatted around the foldout tables and compared results on their phones.

Mr. Cawthorn abruptly conceded the race to Mr. Edwards, shortly after praising Mr. Trump for his endorsement and expressing confidence that the final results would break in his favor to the cheers of supporters. Mr. Edwards said that when the two spoke that night, Mr. Cawthorn offered his full support.

Late Tuesday night on Twitter, Mr. Cawthorn congratulated Mr. Edwards on securing the Republican nomination. He wrote that it was time for Republicans in the district “to rally behind the Republican ticket to defeat the Democrats’ nominee this November.”

Mr. Cawthorn’s spokesman did not respond to a request for additional comment on Wednesday.

Outside Mr. Edwards’s campaign headquarters, people strolling a shop-lined street were unsurprised by his win. Milton Ready, a North Carolina historian who identified himself as an unaffiliated voter who leans Democratic, said he cast a ballot for Mr. Edwards because he seemed like an establishment Republican — but not one obsessed with publicity.

“And,” Mr. Ready added, “I don’t think anyone in the world worries about his sex life or how fast he drives.”