Lonnie Chisenhall crushed a three-run homer and Michael Bourn chipped in a solo shot for the Indians, who avoided a three-game road sweep at the hands of the Rays.
The Tribe banged out 17 hits.
Justin Masterson (2-0) recorded just his second win in 14 career starts against the Rays, hurling seven innings of two-hit ball with eight strikeouts and three walks.
Tampa Bay ace David Price (0-1), meanwhile, experienced his first career loss against Cleveland, surrendering eight runs on 10 hits with three walks over five frames.
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