Lillibridge, whose poor throw contributed to a game-tying Cleveland rally in the ninth, atoned for his miscue with a two-out shot to left off Rafael Perez (4-1), giving Chicago a 7-6 lead.
The Sox then sent five more men to the plate, scoring three additional runs to nail down a win that kept them in the race for the American League Central title. They pulled to within four games of idle first-place Minnesota with the victory.
The Chisox uprising made a winner of Scott Linebrink (2-1), who held the Indians scoreless in the 10th and 11th innings.
Alex Rios went 5-for-6, homered, drove in four runs and scored three times and Paul Konerko chipped in with three RBI for the victors.
Travis Hafner went 4-for-5 with an RBI and Jayson Nix homered for the Tribe.