Brent Morel hit a three-run homer and Paul run seventh inning, leading Chicago 1, in the opener of a four-game series.
Morel finished with two homers and drove in a career-high four runs for the White Sox, who have won four of their last five.
The White Sox announced after the game that infielder Brent Lillibridge suffered a broken hand and will miss the rest of the season. He was hit by a pitch in the seventh inning.
"It was a nice game and we've been swinging the bat real good lately," said Konerko. "But we feel bad for Lillibridge cause he's been having a great year."
"We couldn't get anything going offensively and we didn't match up well," said Cleveland manager Manny Acta. "When their bullpen came out they just shut us down."
Jim Thome posted two hits and knocked in the only run for the Indians, losers of four straight and six of eight.
David Huff (2-4) was charged for three runs on five hits over 6 1/3 innings in the setback.
Leading by a run, Huff ran into trouble in the seventh, starting off the inning by walking Alexei Ramirez and allowing an Alex Rios single. He fanned Tyler Flowers before departing, but Frank Herrmann served up Morel's go-ahead three-run blast to left.
Thornton recorded the final out of the Cleveland seventh, then retired the visitors in order in the eighth. Josh Kinney worked around a leadoff double from Asdrubal Cabrera to turn in a scoreless ninth.
Cleveland picked up a run in the first as Cabrera singled, stole second and scored on a Thome base hit.
Morel's one-out solo shot in the third evened the score.
Morel notched his first career multi-homer game, while Konerko recorded his 10th career grand slam, tying him with Robin Ventura on the franchise's all- time list...Pierre collected his 2,000th career hit in the ChiSox victory, becoming the 268th member of that club.