The crowd of 1,400 people gathered on Saturday at an outdoor amphitheater in Omaha was waiting for Tim Walz, the jocular Minnesota governor running for vice president. But it was another Democrat looking to make history who drew the first standing ovation of the evening.

“Tony, Tony, Tony,” the audience erupted as Tony Vargas, a state senator, stood at the lectern.

“You really know how to make someone feel very, very welcome,” he said.

Mr. Vargas, 40, is vying to become the first Latino to represent Nebraska in Congress, and he has become one of the most notable Democratic candidates on the rise as he competes in a heated House race at the center of national attention.

In 2022, when Mr. Vargas first challenged Representative Don Bacon, a Republican, he lost by less than three percentage points, not even 6,000 votes. Now, their rematch is taking place in the middle of a tightly contested presidential election, and Mr. Bacon’s 2nd District appears to be swinging in Vice President Kamala Harris’s direction, bringing tailwinds for Democratic candidates down the ballot like Mr. Vargas.

Nebraska is solidly Republican, but the 2nd District, which encompasses Omaha and is known as Nebraska’s blue dot, is a swing region that voted for Barack Obama in 2008 and Joseph R. Biden Jr. in 2020. Biden won with 56.4 percent of the vote.

A coveted victory there could help sway control of Congress, as well as decide the next president. Nebraska is one of two states (along with Maine) that award an electoral vote to the winner of each congressional district. That vote could be of such national consequence this year that Republican allies of former President Donald J. Trump in the Nebraska Legislature have been unsuccessfully pushing to change how it is awarded.