Trailing 2-1 entering the frame, Trevor Plouffe drew a lead-off walk from Matt Lindstrom (1-1), went to second on a passed ball and scored on Aaron Hicks' single to tie the game.
One out later, Hicks stole second and Jamey Carroll recorded an infield single.
Donnie Veal relived and issued a walk to Joe Mauer, loading the bases.
Former Twin Jesse Crain then took over the mound and Willingham greeted him by blasting a pitch into the right-center field gap, scoring all three runners.
Minnesota starter Scott Diamond (1-1) earned the decision after yielding two runs (one earned) on four hits over six full frames as the Twins swept a rain-shortened two-game road series.
Glen Perkins turned in a perfect ninth inning to rack up his fifth save.
Adam Dunn cracked a solo homer for the White Sox.
2014: NFL Cheerleaders [PHOTOS]
Michael Sam cut from the Dallas Cowboys