Republicans have elected Senator Thune as the next Senate majority leader, completing a momentous shift in their leadership that elevates a top deputy of Senator McConnell into a key position as President-elect Trump returns to the White House.

Mr. Thune, 63, is in his fourth Senate term representing the state of South Dakota and has promised to work closely with Trump, despite differences the two have had over the years, and will be a crucial part of the incoming president’s efforts to push through his policy agenda.