The Tigers seemed on their way to a 2-1 victory in the ninth but Chicago's Paul Konerko cracked a score-tying homer against Drew Smyly.
The game remained knotted until the 12th, when Miguel Cabrera, making his first appearance since Tuesday due to an abdominal strain, collected a lead-off, pinch-hit single off Dylan Axelrod (3-8).
Pinch-runner Tuiasosopo went to second on Austin Jackson's sacrifice bunt and Hunter followed by sending a game-winning liner to left field, handing the American League Central-leading Tigers a four-game sweep of the Chisox.
Bruce Rondon (1-1) earned the decision after Tigers starter Rick Porcello had allowed just one run on eight hits over 7 2/3 innings.
Adam Dunn went 4-for-5 with a solo homer for reeling Chicago, which dropped a 10th consecutive game.
Swim Week Miami Beach 2014 [PHOTOS]