CROWLEY, La. — In mid-December last year, Capt. Clay Higgins of the St. Landry Parish Sheriff’s Office was standing before a camera for a Crime Stoppers video, lecturing three at-large suspects in the thefts of several four-wheelers.

“The families that have been hit all have one thing in common: They work for a living,” he said in the cadenced monotone of a disgruntled football coach. “The three of you just sit around and whine about how unfair life has been for you, and then you suck on your pipe to escape the disgraceful reality of your own existence. The three of you together are not worth the expense of one bullet.”

In mid-December of this year, Mr. Higgins won a seat in Congress.

The victory, in a Dec. 10 runoff, was at once shocking and almost predictable: It represented both a stunning upset by a political novice with a baggage train of alleged transgressions against a veteran officeholder and favorite son in Cajun country, and an entirely natural outcome for an election season like this one.

While anti-establishment outsiders have long competed in congressional races, they have typically come up short. Veteran politicians are seen as safer bets against partisan opponents in a general election. But Mr. Higgins benefited from Louisiana’s curious election system — a nonpartisan primary followed by a top-two runoff. So he was running against a fellow Republican after voters had already seen that a rough-edged, anti-establishment outsider — Donald J. Trump — could actually win.