That brought up Escobar, pinch-hitting for Brent Lillibridge. With the Milwaukee outfielders drawn in, he delivered a sharp hit into the left-center field gap for the game-winner.
The victory gave the White Sox the rubber match of a three-game interleague series and, combined with Cleveland's loss to Houston, vaulted them over the Indians into first place in the American League Central.
Brian Bruney (1-0) notched the win with a scoreless top of the 10th after Chisox starter Jose Quintana had tossed eight scoreless frames.
Michael Fiers was saddled with a no-decision for Milwaukee after hurling 7 1/3 scoreless innings, recording seven strikeouts.