John Kennedy, the Republican state treasurer of Louisiana, won the state’s U.S. Senate race on Saturday, giving Republicans a 52-seat majority in the new Senate next year.

With 32% of the vote counted, Mr. Kennedy was declared the winner over his Democratic rival Foster Campbell, a cattle farmer and state utility regulator, 63% to 37%, in a special runoff election triggered by Louisiana’s unique election rules.

The win by Mr. Kennedy helps solidify GOP control of the Senate when the new Congress convenes in January. But that still leaves a slim margin for Senate Majority Leader Mitch McConnell (R., Ky.), who will be responsible for passing much of President-elect Donald Trump’s ambitious agenda through a closely divided chamber.

The Democrats will have 46 senators when the new members are sworn in Jan. 3, along with two independents, Angus King of Maine and Bernie Sanders of Vermont, who caucus with them.

Meanwhile, two vulnerable Democratic members, Heidi Heitkamp of North Dakota and Joe Manchin of West Virginia, are being considered positions in the Trump administration. Their move to a Trump administration post could give Republicans additional seats given the competitive politics in both states.