Matt LaPorta's RBI double in the eighth inning proved to be the difference, as Cleveland rallied past Minnesota, 3-2, in the opener of a three-game set.
Carlos Santana added two hits, including a run-scoring double, for the Indians, who have won three of four.
Rafael Perez (5-2) recorded the final out in the eighth to earn the win, and Chris Perez turned in a perfect ninth to record his 24th save.
Tsuyoshi Nishioka collected a pair of hits and drove in both runs for the Twins, losers in seven of their last eight.
Glen Perkins (4-3) allowed the winning run over two-thirds of an inning to take the loss.