Dunn's blast in the third tied the game 2-2 and his run-scoring two-bagger in the fifth gave the Chisox a 4-2 advantage.
The designated hitter capped his big day with another two-run clout in the eighth as Chicago banged out 14 hits in the rubber match of a three-game series.
Alex Rios went 3-for-4 with three runs scored and Dayan Viciedo provided a solo blast with two RBI for the winners.
Dylan Axelrod (1-3) earned his first victory over the season in his eighth start, allowing three runs on nine hits in a 5 1/3-inning effort. The right-hander yielded one walk and struck out four.
Twins starter Mike Pelfrey (3-4) was tagged for five runs on eight hits over four full frames.
Eduardo Escobar homered and Jamey Carroll added three hits, including an RBI single, in Minnesota's loss.
The Twins' Joe Mauer went 3-for-5, raising his batting average to .349