Cabrera's solo blast in the fourth inning broke a 2-2 tie and Kosuke Fukudome added a bases-empty clout in the next frame for the Indians, who also took the first game 4-3.
Travis Hafner went 2-for-4 with a with a two-run homer for Cleveland in the second game.
Fausto Carmona (7-15) lasted six innings to earn the win, allowing three runs on seven hits while walking a pair and striking out three.
Chicago's Gavin Floyd (12-12) surrendered four runs on seven hits through 6 2/3 innings.
Alejandro De Aza drove in a pair of runs for the White Sox.
Swim Week Miami Beach 2014 [PHOTOS]