It was Swisher's fourth homer of the season and first since being activated from the disabled list on Thursday before the series opener. For the second straight day, Cleveland won after trailing Boston 2-1 in the seventh.
The Red Sox wasted an excellent opportunity to win the game in the ninth when they loaded the bases with three walks by John Axford. But Brock Holt ended the threat with a groundout to second against reliever Scott Atchison.
Cody Allen (3-1) ended the game with two perfect innings, striking out three.
Junichi Tazawa (1-1) took the loss one day after walking in the winning run in Cleveland's 3-2 victory.
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