Peavy (11-5), obtained from the White Sox in a July 31 trade-deadline deal, held his ex-club to five hits and a walk while striking out four.
It was the sixth win in seven games for American League East-leading Boston, which came into the contest with a 3 1/2-game bulge over second-place Tampa Bay.
Chicago's John Danks (4-11) was battered for six runs (five earned) on 11 hits over five innings.
Alejandro De Aza and Avisail Garcia each provided an RBI for the Chisox.
Toddler uninjured after being knocked over by Obama family dog
Man spent 15 hours in jail for plugging electric car into an outlet at a school