The Friday before Election Day, Representative Ruben Gallego invited his supporters to Rancho Ochoa, a rodeo venue in southwestern Phoenix, where they listened to a brass band, cheered as bull riders strained to hold on and watched the dancing horses, a staple of Mexican rodeos.

It was the celebratory culmination of an extensive effort by Mr. Gallego, a Democrat, to target Latino voters as he vied for Arizona’s open Senate seat — an effort that appeared to pay dividends. Even as Latinos, and especially Latino men, shifted drastically away from Democrats this year, and President-elect Donald J. Trump beat Vice President Kamala Harris in Arizona by more than five percentage points, Mr. Gallego won his race by over two points. He seemed to outperform Ms. Harris with Latinos easily.

The shift toward Mr. Trump left Democrats, who have long operated with a belief that demographic change would equal a winning destiny, stunned and scrambling for answers. Some are looking to Mr. Gallego, a plain-talking military veteran and the son of Colombian and Mexican immigrants who has plenty of thoughts on how his party can win back working-class voters and avoid taking the Latino community for granted.

In an interview, Mr. Gallego said the Democratic Party had failed to address the deep-seated anxiety that Latino men felt over rising prices, which left them unable to provide for their families no matter how much harder they worked.

“Latino men feel like their job is to provide security for their family — economic security and physical security,” he said. “And when that is compromised, they start looking around.”