Lonnie Chisenhall added a two-run hit for the Indians, while Michael Bourn reached base three times on two singles and a walk and scored twice.
Justin Masterson (3-4) fanned 10 over seven scoreless frames, giving up four walks and three hits for the win. Cody Allen tossed a perfect ninth for his fourth save.
Xander Bogaerts hit a two-run home run for the Red Sox, who had recorded sweeps of Atlanta and Tampa Bay in the last week to erase the memory of a 10- game slide.
John Lackey (6-4) went the distance in defeat, allowing eight hits and three runs.