Dunn's two-run shot in the seventh inning erased a 4-2 Minnesota lead and the White Sox went on to score seven more runs, including two on Dunn's eighth-inning double.
Chicago is tied with Detroit atop the American League Central, following the Tigers' loss to Cleveland earlier Tuesday.
Nate Jones (4-0) was credited with the win thanks to one scoreless inning in relief of starter Jose Quintana.
Twins starter Cole De Vries surrendered one earned run over six innings, but Casey Fien (1-1) took the loss as Minnesota's bullpen was lit up for nine runs.
Josh Willingham drove in all four Minnesota runs with a pair of homers, his 25th and 26th of the year, as the Twins lost for the fifth time in seven games.