Democrats have held majority control of New Jersey’s Legislature for 23 years. The state has had a Democratic governor since 2018. And Democrats enjoy an 800,000-voter registration advantage.

So when Republicans talk about November’s race for governor, they acknowledge that their sense of growing optimism can feel a bit unfamiliar.

“It’s viewed, nationally, as not impossible,” said William Palatucci, one of New Jersey’s two representatives on the Republican National Committee. “And there’s a lot of interest for that reason.”

On Tuesday, Jack Ciattarelli, a former state assemblyman, won the Republican nomination by a resounding 3-to-1 margin, further ratcheting up confidence among party leaders intent on making the most of an atmosphere that they believe offers the best shot for a win in years. He will run against Representative Mikie Sherrill, a Democrat who beat five opponents, decisively, after a bruising and expensive contest.

“It’s the best opportunity, maybe, in a generation,” said Mike DuHaime, a political strategist who helped to run campaigns for former Gov. Chris Christie, a Republican who left office in 2018.

Several factors are expected to boost Mr. Ciattarelli’s odds.

After competing for governor in 2017 and 2021, he has a sturdy level of name recognition and a deep understanding of the policy issues that are likeliest to motivate voters. He also has the world’s biggest cheerleader — President Trump — in his corner and history on his side. Not since 1961 has either party managed to hold on to the governor’s office for three consecutive terms, and Gov. Philip D. Murphy, a Democrat, has had the job for two.