Kazmir (10-9) allowed one run on six hits and walked two in a 104-pitch performance that gave the surging Indians a ninth straight win and a one-game lead in the race for the first American League wild-card spot.
The Indians could wrap up a post-season berth as well as clinch home-field advantage for Wednesday's one-game AL wild-card play-in with another win over Minnesota Sunday.
Carlos Santana broke a scoreless tie with a two-run homer in the fourth inning and Cleveland added three more in the fifth.
The Tribe waited out a 2-hour, 41-minute rain delay at Target Field in Minneapolis before recording the clutch win.
The loss was tagged on Minnesota's Cole DeVries (0-2), who gave up five runs on six hits in five innings.
Eric Fryer supplied the only RBI for the Twins, who dropped a fifth straight.