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.
Wisconsin business offering 'therapeutic cuddling' forced to close
Reindeer recovered after escaping from Santa during lighting ceremony