Senator Darline Graham won the Republican Senate nomination in South Carolina on Tuesday, relying on the strength of President Trump’s political capital to overcome her inexperience as a first-time candidate for public office.

Ms. Graham defeated Representative Ralph Norman, a five-term congressman, in deeply red South Carolina after Mr. Trump did more to support her than he had for any other Republican candidate this year — out of apparent affection for her late brother, Lindsey Graham, who died last month.

Her win showed the president’s continued ability to move Republican primary voters, especially when he goes all in on a race. His chosen candidates won a series of key primaries in the spring, but a few of his picks have more recently lost, putting the power of his endorsement in question. That he could lift Ms. Graham, who is poised to win a six-year Senate term after being virtually unknown six weeks ago, may have settled that question, for now.

Trump’s pick also prevailed in Oklahoma’s Republican primary for governor, where Mike Mazzei, a former state lawmaker, defeated Gentner Drummond, the state’s attorney general. Mr. Mazzei was ahead by less than half of a percentage point when The Associated Press called the race.

Here are six takeaways from Tuesday night’s primaries:

Trump’s support faced a big test in South Carolina. He aced it.

Mr. Trump won South Carolina by more than 18 percentage points in 2024. But twice in the past six months, candidates he endorsed have fallen in the state.

Ms. Graham looked like she could become one of only a handful of Trump-backed Senate candidates to lose their primaries nationwide since 2017. She initially struggled to introduce herself to voters in the extraordinarily short campaign and flubbed a debate answer about national security.

But on the campaign trail, she said, voters asked her about kitchen-table issues, not national security. And she wrapped her candidacy around her support for Mr. Trump and his for her. She will now face Dr. Annie Andrews, a pediatrician and the Democratic nominee, in the November general election, where Ms. Graham will be heavily favored.

Trump-backed candidate prevailed in a tough fight in Oklahoma.

Mr. Mazzei, the president’s pick to succeed Oklahoma’s term-limited governor, Kevin Stitt, won the Republican nomination after a neck-and-neck race against Mr. Drummond, a former Air Force pilot who is in his first term as attorney general.

Mr. Trump not only endorsed Mr. Mazzei but also went out of his way to attack Mr. Drummond during the campaign.

“Everybody I know dislikes him,” Mr. Trump told supporters in a telephone rally for Mr. Mazzei on Monday.

A key issue in the race was a $4 billion aluminum smelter project that has been a priority of Mr. Trump’s. Mr. Mazzei initially opposed it and then reversed his position. Mr. Drummond campaigned against it.

Where Trump goes, his voters vote.

Mr. Trump rallied for Ms. Graham last Friday in Myrtle Beach, an area where the candidate who came in third place in the primary, Representative Russell Fry, performed best. On Tuesday, Ms. Graham trounced Mr. Norman in Horry County, home to Myrtle Beach.

Ms. Graham also did well across the eastern part of South Carolina, including in Charleston County, which had been a base of support for another rival, Mark Sanford, in the primary.

When Trump’s allies send texts, his voters vote.

Mr. Trump’s super PAC, MAGA Inc., spent more than $800,000 on the special primary runoff, nearly a third of the organization’s total spending in this election cycle. And South Carolina Republicans had the evidence on their cellphones: They said they received repeated text messages, including on Tuesday, urging them to cast ballots for Ms. Graham.

Republicans nationwide have fretted about MAGA Inc. not spending as quickly as some in the party would like to help vulnerable Republicans. The organization’s investment in South Carolina may not stop the grumbling, given the ongoing battle for control of Congress and the fact that the Senate seat was expected to stay in Republican hands regardless of who won the nomination. But it will allow MAGA Inc. to say that when it does enter a race, it does so decisively.

Winning South Carolina’s most conservative region was not enough for Ralph Norman.

Mr. Norman performed best in the Upstate region of northwest South Carolina, which includes Greenville and Spartanburg and surrounds his congressional district. The area is rich in Republican voters who lifted him into second place in the crowded primary two weeks ago.

But the rest of South Carolina went for Ms. Graham, leaving Mr. Norman without the broader support needed to win a two-way runoff. His supporters pinned much of their hopes on Tuesday on Greenville County. But they were not enough.

Mr. Norman told supporters in Rock Hill, S.C., on Tuesday night that he “couldn’t have done anything different,” but he did lament that Mr. Trump took sides in the race.

“He has that right,” he said, as his supporters booed. “He has that prerogative.”

Lindsey Graham’s legacy didn’t sink his sister.

Mr. Graham, who was running for re-election when he died, won his Republican Senate primary in June with 56 percent of the vote. That was a low ceiling for a four-term incumbent, and one he reached only after he and his allies spent more than $18 million on the race. He had long struggled with some Republican voters who were skeptical of his past support for immigration reform and for Mr. Trump’s rivals in the 2016 presidential primary.

When Ms. Graham entered the race to succeed him, her ceiling was even lower: She was a first-time candidate known to voters only as the sister of a senator they had complicated feelings about. And many Republican voters said they disliked the perception of nepotism.

Ms. Graham said she knew she had to earn voters’ support. She tried to break with her brother, saying she was more focused on domestic policy and would take a harder line on immigration.

“I continue to miss Lindsey every day, and there is still a big hole in my heart,” she told supporters in Columbia, S.C., on Tuesday night. “But I know he’s looking down right now very proud of everyone in this room.”

Time to brush up on foreign policy.

Reporting was contributed by Emily Davies in Columbia, S.C.; Eduardo Medina in Myrtle Beach, S.C.; Chris Cameron in Washington, and Caroline Soler and Shane Goldmacher in New York.