Michael Bourn doubled to score Lonnie Chisenhall and Drew Stubbs to give the Indians a 2-0 lead in the fourth inning.
Chad Allen pitched a scoreless seventh for the Tribe, but Joe Smith (4-1) surrendered the winning run in the eighth on an RBI single by Joe Mauer.
Tribe hitters struck out 11 times against starter Mike Pelfrey and four relievers.
Despite the loss, the Indians are still only 1.5 games out of first place, as the Tigers lost to Kansas City 1-0.
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