Representative Andy Kim, a lawmaker who has turned New Jersey politics on its head since entering the race to unseat Senator Robert Menendez, won the Democratic nomination for Senate on Tuesday after a campaign marked by a watershed ballot-access ruling.

The victory makes Mr. Kim, 41, a favorite to become New Jersey’s next senator. He would be the first Korean American to be elected to the U.S. Senate.

“I took the chance to run for Senate eight months ago on the belief that people are fed up with our broken politics and are ready for a new generation of leadership,” Mr. Kim said in a statement. “What I found is that there is a deep hunger across the political spectrum for a different kind of politics grounded in integrity and public service that aims to rebuild trust.”

The results, announced by The Associated Press minutes after polls closed, capped a tumultuous campaign that began a day after Mr. Menendez, a Democrat, was accused in September of being at the center of a sprawling international bribery scheme. Mr. Menendez, a third-term senator who for decades shrewdly navigated New Jersey’s notoriously rough-and-tumble politics, did not compete for the Democratic nomination.

For months, Mr. Kim’s main opponent had been Tammy Murphy, the wife of Gov. Philip D. Murphy. Ms. Murphy, a first-time candidate, was bombarded by accusations of nepotism and dropped out of the race in March after failing to connect with voters.

Mr. Kim, a former member of the National Security Council who advised President Barack Obama on Iraq, was first elected to Congress in 2018 — one of four Democrats that year to flip a New Jersey district from red to blue after Donald J. Trump was elected president.