Gordon Beckham posted three doubles, an RBI and a run scored for Chicago, which won for the second time in three games.
"It was one of those days I've searching for," Beckham said. "I'm glad we came back and won."
Matt Thornton (2-5) earned the win with 1 1/3 scoreless innings of relief. Dylan Axelrod allowed four runs on seven hits and four walks through 4 2/3 innings in his second major-league start.
Zach Putnam (0-1) allowed two runs while recording one out in the seventh for the loss. Tribe starter Zach McAllister lasted 5 1/3 innings, allowing two runs on six hits and a walk in his third career start.
Jason Donald went 4-for-5 with a run scored for the Indians, losers in two of their last three.
After back-to-back singles by Donald and Carlos Santana to start the fourth, Jim Thome pulled a ball through the right side of the infield, scoring Donald and moving Santana to third. Shelley Duncan then walked to load the bases and Grady Sizemore's sacrifice fly to left brought Santana across the plate.
Matt LaPorta stepped up next and hit a first-pitch double over the head of Dayan Viciedo in right field to bring home Thome. Jack Hannahan's sacrifice fly to center scored Duncan to make it a 4-0 game.
The White Sox answered with a run in the fourth. Alejandro de Aza singled, moved to second on Omar Vizquel's base hit and came home on Beckham's two-out double high off the wall in left field.
Perez gave a free pass to Adam Dunn, but then got Viciedo to hit a grounder to short. Donald's throw to second was a little too far to the right, making Luis Valbuena's relay throw to first late and Viciedo reached safely. De Aza followed with a double to left to plate Ramirez and Viciedo, bringing the White Sox within one.
"We lost the game in the sixth inning," said Indians manager Manny Acta. "If we turn that double play it's a different story."
Putnam started the seventh for Cleveland and was greeted by a Tyler Flowers base hit and a Beckham double to put runners on second and third. Pierre brought Flowers home with a groundout to second and Ramirez's run-scoring, bloop single to shallow center field gave Chicago a 5-4 lead.
Cabrera sat out the game after straining his back on a swing in the first game of the doubleheader...Chicago leads the season series 10-6...The White Sox have recorded an extra-base hit in a season-high 36 straight games...Chicago went 3-for-11 with runners in scoring position and left seven men on base...Cleveland went 2-for-7 with runners in scoring position and stranded eight.