In the contest for governor, Mr. Trump endorsed Ms. Lake, a telegenic former newscaster who had become an unabashed champion of Trumpism. Mr. Trump is seeking some redemption after struggling earlier this year in other governor’s races, most notably failing in his attempt to oust the Republican governor of Georgia, Brian Kemp.

Unlike Mr. Kemp, the Republican governor of Arizona, Doug Ducey, who earned Mr. Trump’s ire by supporting the results of the 2020 election, was term-limited and not on the ballot himself. Mr. Ducey put his support behind Karrin Taylor Robson, a wealthy real estate developer who spent more than $18 million on her run and who also had the backing of Mike Pence, Mr. Trump’s former vice president.

Ms. Lake, who has made voter fraud a centerpiece of her candidacy, declared victory at a moment when she was actually behind in the vote counting. “We won this race,” Ms. Lake said at her election-night party. “Period.” She later took the lead for the first time, but that contest remained too close to call.