Recalled from Double-A Frisco prior to the game, Texas starter Scott Feldman (4-3) gave up one run on four hits in six innings to pick up the win.
Hank Blalock hit a two-run homer for the Rangers, who moved into second place in the AL West.
Javier Vazquez (7-8) allowed three runs on four hits with 10 strikeouts over seven innings to take the loss for Chicago, which has lost four of five.
Boone Logan gave up three runs in one-plus innings of relief. Carlos Quentin homered for the White Sox, who remain a half-game ahead of the Minnesota Twins for the top spot in the AL Central.