The Royals took a three-run lead into the ninth before Broxton ran into trouble on his way to his 20th save.
The Kansas City closer got Justin Morneau on a fly to center that sent Mauer to third and Trevor Plouffe popped up to third base to end the threat.
Luis Mendoza (3-4) claimed the win after working eight innings in a game for the first time in his career. He gave up five hits and a run without walking a batter. He had not won since May 13.
Kansas City won its fourth in a row, with Salvador Perez hitting a solo homer in the second and Yuniesky Betancourt posting two hits and an RBI.
Brian Duensing (1-4) surrendered three runs on eight hits in four-plus innings to take the loss, the Twins' third straight.