COLUMBUS, Ohio—J.D. Vance, an author and venture capitalist, won Ohio’s Republican primary for U.S. Senate in a victory that demonstrated former President Donald Trump’s influence over GOP voters.

Mr. Vance, 37 years old, was declared the winner on Tuesday by the Associated Press in a seven-candidate race that had showcased the combative brand of conservatism Mr. Trump helped usher into the party. He benefited from millions of dollars in advertising funded by billionaire Peter Thiel, a technology entrepreneur.