Quintana (4-1) allowed one run on two hits and one walk for the White Sox, who have won seven of their last nine games.
Addison Reed picked up his 12th save with a scoreless ninth and Kevin Youkilis had a home run and scored two runs for Chicago.
Rangers' starter Matt Harrison (11-4) went eight innings, allowing two runs on eight hits and two walks. The loss broke a seven-decision winning streak for Harrison.
Adrian Beltre drove in a run for Texas, which has lost four straight.