Ron Hassey's home run with one out in the 11th inning in the second game in Boston gave Cleveland a sweep of the doubleheader.
In the opener, Ed Whitson continued his mastery over the Red Sox with seven innings of scoreless pitching and Rick Manning drove in two runs with a pair of singles.
Hassey's homer, his fifth of the season and first at Fenway Park, came off Luis Aponte, who entered the game to start the inning. The hit made a winner of reliever Tom Brennan (2-2), who entered the game in the 10th.
The Red Sox put runners on first and third with none out in the 11th when Carney Lansford doubled and moved to third on Wade Boggs' single, his fourth hit of the game, raising his average to .378. However, the Indians got out of the jam when Brennan caught Reid Nichols' bunt and doubled Lansford off third base. Brennan then struck out Rick Miller to end the game.
Whitson, who shut out Boston on seven hits five days ago, allowed seven hits before giving way to Dan Spillner after injuring his left knee. Whitson, who struck out three and walked one, has now pitched 20 consecutive scoreless innings.