In Boston, Rich Gedman had three hits, including a double in the fifth that knocked in the game-winning run, and Carl Yastrzemski hit a two-run homer.
Gedman's double, off Wayne Garland (3-7), scored Carney Lansford, who had doubled.
Bob Stanley (10-7) relieved starter Bruce Hurst in the fourth and gave up an RBI single to Tom Veryzer but held the Indians hitless thereafter.
Yastrzemski gave Boston a 2-0 lead against Garland in the first when he homered into the right field stands. It was his seventh homer this year and 426th of his career, tying Billy Williams for 16th place on the all-time list.
The Indians scored in the second when Bo Diaz doubled, went to third on a wild pitch and scored on Andre Thornton's double-play grounder.
The Red Sox made it 3-1 in the third when Dwight Evans singled, moved to second on an infield out and scored when Lansford singled. Cleveland went ahead, 4-3, in the fourth on an RBI double by Alan Bannister, Rick Manning's grounder and Veryzer's single.
Evans tied the score, 4-4, in the fourth when he singled home Gedman with his 1,000th major league hit.