Peavy (9-7) allowed single runs in each of the first two innings but then shut the door, retiring 15 of the next 16 Twins batters en route to the win.
The right-hander ended up yielding a pair of runs (one earned) on five hits and didn't issue a walk as the American League Central-leading White Sox took the rubber match of a three-game series.
Addison Reed picked up his 18th save despite allowing a pair of runners to reach base in the ninth.
Minnesota's Scott Diamond (9-5) pitched well but absorbed the loss, surrendering three runs on eight hits over 7 1/3 frames.
Danny Valencia socked a solo homer for the Twins.
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