Peavy (10-5), making his fifth start for Boston since being obtained in a trade-deadline deal July 31, broke a personal three-start winless streak in the 111-pitch gem.
The veteran right-hander allowed one run, issued one walk and struck out five as the Red Sox took the rubber match of a three-game series between two likely post-season contenders.
Mike Napoli belted a two-run homer and finished with three RBI while Shane Victorino and Jarrod Saltalamacchia each contributed a home run for the Red Sox, who forged a one-game lead over second-place Tampa Bay the American League East.
Dustin Pedroia went 3-for-4 with an RBI for Boston.
Chris Capuano (4-7) yielded three runs on six hits in a five-inning performance for the NL West-leading Dodgers, who dropped a series for the first time since June 14-16.
Adrian Gonzalez homered for Los Angeles.