With the score tied 6-6, Minnesota's Danny Valencia led off the ninth with a single off Brett Myers (0-1) and was replaced on the base paths by pinch-runner Casilla. The next batter, Brian Dozier, laid down a bunt that White Sox catcher A.J. Pierzynski fielded but threw wildly to first base, allowing Casilla to reach third.
Carroll followed by lofting a fly ball to right, sending home the game-winner and handing the Twins a victory in the opener of a three-game series.
Valencia finished 2-for-3 with a pair of RBI and Ryan Doumit knocked in a pair for Minnesota, which claimed its fourth straight win.
Glen Perkins (2-1) was credited with the victory thanks to a scoreless top of the ninth.
Alejandro De Aza went 4-for-5 with a pair of run-scoring singles, including the tying run in the eighth. Pierzynski bashed a two-run homer for the American League Central-leading White Sox.