WEST PALM BEACH, Fla.—Republicans shored up their U.S. Senate majority on Sunday when Florida Gov. Rick Scott narrowly prevailed over Democratic Sen. Bill Nelson, who conceded defeat nearly two weeks after Election Day as a second recount left him trailing by about 10,000 votes.

Mr. Scott’s pickup of a Democratic-held seat gives the GOP a 52-to-47 edge in the Senate, with one seat still to be decided by a runoff election in Mississippi this month that the Republican is favored to win. That would give the GOP a net gain of...