After sending Chris Sale to his first loss after a 9-0 start, the Indians got three runs and five hits in six innings against Joel Quintana (5-4). Quintana's ERA, an AL-best 1.98 at the start of the day, rose to 2.22.
Before Tuesday, the White Sox had been 15-3 in games started by Sale and Quintana.
Cleveland, which trailed by six games before play on May 10, closed within a half-game of the AL Central-leading White Sox, who lost for the ninth time in 12 games. Chicago finished a 3-7 homestand in which it scored more than three runs just twice.
