Quintana (2-1) managed to work around 10 hits, surrendering only one run over 5 1/3 innings. The 23-year-old Colombian didn't issue a walk and struck out four.
The American League Central-leading White Sox claimed the opener of a three-game interleague series.
Adam Wainwright (5-7) was saddled with the loss despite issuing only two runs on five hits over seven frames.
Allen Craig drove in the Cardinals' only run.
2014: NFL Cheerleaders [PHOTOS]
Michael Sam cut from the Dallas Cowboys