With a decisive margin in the Senate, Republicans, who have shown their willingness to accommodate the president-elect, will have the numbers to overcome divisions over his personnel and policies.
Tim Sheehy, a wealthy Republican businessman, was dogged by controversies during the campaign. But Senator Jon Tester, a Democrat, could not outrun the demographic shift in his deep-red state.
After picking up seats in West Virginia, Ohio and Montana and winning an unexpectedly close race in Nebraska, the G.O.P. sealed a majority. Tight races in swing states will determine their margin.
The new majority will be under new leadership for the first time in 18 years since Senator Mitch McConnell, Republican of Kentucky, is stepping down from the top post he has held since 2007.
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