Nathaniel Lowe hit an RBI double off Luke Weaver (3-4) in the seventh to give Boston a 4-3 lead.
Boston (69-59) now sits just half-of-a-game behind New York (69-58) for the top Wild Card spot in the American League. The Red Sox win also spoiled a joyous night for Yankees' fans, who were celebrating the second annual "Seinfeld Night", where the first 18,000 fans received a bobblehead of George Costanza sleeping under his desk.
Anthony, who had an RBI single in the sixth, hit his fifth career homer when he connected off Yerry De Los Santos after first baseman Paul Goldschmidt committed New York's fourth error.
The Red Sox went 3 for 19 with runners in scoring position, snapped a three-game losing streak and moved within one game of the Yankees for the AL's first wild-card spot.
Ceddanne Rafaela scored Boston's first run on a throwing error by catcher Ben Rice.
Rice homered and Goldschmidt hit an RBI single for the Yankees, whose fourth five-game winning streak was stopped and who lost for the fourth time in their past 13 games.
After Boston starter Lucas Giolito allowed three runs and five hits in 4 2/3 innings, five relievers combined on 5 1/3 scoreless innings, including Greg Weissert (5-4) who held the Yankees to one hit in 1 1/3 innings.
New York starter Luis Gil allowed two runs -- one earned -- and four hits in five innings. He issued five walks in his fourth start since returning from a lat strain.
Steven Matz allowed a triple to Rice in the seventh before retiring Goldschmidt to protect the one-run lead. Weissert kept the game tied in the sixth and Aroldis Chapman retired Giancarlo Stanton for his 23rd save.
The Red Sox are 6 for 47 with runners in scoring position in their past four games.