BATON ROUGE, La.—Democrat John Bel Edwards pulled off an improbable win in the Louisiana governor’s race on Saturday, by running as a centrist who could right the state’s fiscal ship and by capitalizing on the unpopularity of Republican Sen. David Vitter.

In a Deep South state that has been solidly Republican, Mr. Edwards received 54.8% of the votes cast in the gubernatorial runoff election with 89% of the precincts counted.

Mr....