Batting out of the seventh spot, Brent Lillibridge, who played for both the Red Sox and White Sox earlier this season, made his mark with the Indians by going 3-for-4 scoring twice and driving in a pair of runs.
In his 11th game as an Indian, Lillibridge muscled out his first home run of the season to give Cleveland a 1-0 lead in the second, then tied the game at two with an RBI single in the fifth.
After Cleveland took a 3-2 lead in the sixth, he led off the seventh with a double and came home on a sac fly, adding an insurance run.
The loss sent the Red Sox 12 games behind the Yankees in the AL East, while Cleveland's victory still left them nine games behind front-running Chicago in the Central.
While the Braves have to be content for now to trail the Nats by 4 1/2, they still lead the wild card race by three games over Pittsburgh, thanks largely to Freddie Freeman, who doubled home a pair of runs in the opening frame, then capped a seven run Atlanta outburst in the seventh with a three-run homer, his 14th of the season.
Freeman accounted for five of Atlanta's nine runs, upping his RBI total to 74, tying him with three other players for sixth place in the category. Carlos Beltran leads the RBI race with 82.
Looking for a sweep of their three-game series in New York, the Braves square off with the Mets at 8:05 pm EDT in ESPN's featured Sunday night game.