Slowey (12-6), making his second start after returning from the disabled list with a sore elbow, lasted five innings and yielded only two unearned runs on six hits.
Glen Perkins, Jose Mijares, Jon Rauch and Brian Fuentes combined to toss four shutout innings to help the Twins win the rubber match of a three-game set against the Indians and take their eighth victory in nine games.
Minnesota entered the contest with a six-game lead over the second-place Chicago White Sox in the American League Central.
Michael Cuddyer collected a pair of hits and an RBI to lead Minnesota at the plate.
Cleveland starter Mitch Talbot (9-12) left the game with an injury after only three batters. Justin Masterson came in and surrendered allowed three runs (one earned) in seven innings of relief.