The White Sox rebounded from an 8-6 loss in the first game, which was scheduled to make up for a May 28 rainout. Mark Teixeira hit two homers and six RBI in the opener.
Brandon McCarthy (1-1), who had a no-hitter through 4 1/3 frames, gave up just two hits in 7 2/3 innings to end the Rangers' three-game winning streak.
Dye went 3-for-4 and had six RBI. Scott Podsednik also had three hits and scored twice.
Edison Volquez (0-1) took the loss for Texas in his major league debut, allowing five runs in 4 2/3 innings.