Being mocked on “Saturday Night Live” didn’t hurt—and may have helped— Texas Republican Dan Crenshaw, a retired Navy SEAL and political newcomer who easily won Tuesday his House race.

With 87 percent of precincts reporting, Mr. Crenshaw was declared the winner with 53 percent of the vote, besting Democrat Todd Litton, who had drawn 45 percent.

Mr. Crenshaw was favored to win the Houston-area seat being vacated by retiring Republican Rep. Ted Poe even before last weekend’s “SNL” episode mocking his appearance spurred a national backlash.

Cast member Pete Davidson said during a segment that the candidate looked like “a hitman in a porno movie,” referring to the black eye patch Mr. Crenshaw wears after losing an eye in Afghanistan.

The joke was widely denounced as tasteless and insensitive. A number of Republicans called on Mr. Davidson and the show’s producers to apologize, which they have not done.

At his victory party Tuesday in Houston, Mr. Crenshaw brushed off the episode, saying, “I’m from the SEAL team so we don’t really get offended,” according to the Houston Chronicle.

Mr. Crenshaw said Monday he was uninterested in “some hollow apology,” calling instead for those affiliated with the long-running NBC show to donate to veterans’ groups.

“There’s a lot of veterans that really need help, and frankly, this kind of thing is offensive to them,” Mr. Crenshaw said on CNN. “They feel laughed at.”

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