Trailing 3-2 entering the frame, Alexi Casilla led off with a single off Indians starter Jeanmar Gomez. Rafael Perez (2-1) came in to pitch and Denard Span reached when second baseman Orlando Cabrera fumbled his potential double-play grounder.
Kubel made the Tribe pay by bashing a pitch to right-center that hit high off the wall, chasing home both runners.
Glen Perkins, Jim Hoey and Jose Mijares combined for a scoreless eighth inning and Matt Capps made the lead hold up, gaining his fourth save in the ninth as Minnesota swept a rain-abbreviated two-game series with Cleveland.
Twins starter Carl Pavano (2-2) earned the victory after yielding three runs on six hits and two walks over seven innings. He struck out three.
Michael Brantley smacked a two-run double for the Indians. The blast was originally called a home run but was ruled not to have left the ballpark after a video review.