The Tigers, after waiting out a rain delay, were trailing 8-6 going into their final at-bats, but Austin Jackson hit a one-out triple off Sergio Santos (3-4) and Raburn followed with a two-run smash to tie the score.
After Delmon Young struck out, Cabrera ended it with a solo shot and the American League Central-leading Tigers took the middle contest of a three-game set.
Luis Marte (1-0) earned the win with a scoreless top of the ninth for Detroit, which won for the fourth time in five games.
Tigers starter Brad Penny coughed up eight runs (four earned) on 10 hits and a walk in five innings.
Chicago's Gavin Floyd was touched for four runs on eight hits through five innings, walking no one and striking out four.
Alexei Ramirez contributed a three-run homer for the White Sox, who dropped 7 1/2 games behind Detroit in the American League Central.