Peavy (5-1) logged his first win in four starts, shutting down the Cubs on three hits while walking two and striking out seven.
The right-hander tossed 70 of his 100 pitches for strikes and lowered his ERA to 2.39 as the White Sox completed their first road sweep of the Cubs since 1999 and won a season-high fourth straight contest.
Cubs starter Paul Maholm (4-3) absorbed the loss, surrendering five runs on nine hits over 6 1/3 frames.
The Cubs dropped their sixth straight game.
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