Quintana (6-3) allowed a second-inning single to Rodriguez in his first at-bat of the season following hip surgery and a strained quadricep, but otherwise controlled the Yanks on one run, allowing eight hits in all.
The left-hander walked one and struck out five as the White Sox snapped a 10-game in losing streak in the opener of a three-game set.
Yankees starter Andy Pettitte (7-9) was lit up for seven runs on 11 hits over just 2 2/3 innings.
Rodriguez, who was playing while appealing a 211-game suspension for performance-enhancing drug use handed down before the game, was roundly booed by the White Sox fans during his introduction.
Batting cleanup, he finished the game 1-for-4.
Brett Gardner drove in the only run for New York, which fell for the sixth time in nine games.