Paul Konerko led off the 10th in a 3-3 tie game with a single against Chris Perez (0-1). One out later, pinch-runner Brent Lillibridge came around to score when Rios lined a pitch into the right-center field alley for a three-bagger.
Rios then came home to score an insurance run on Alexi Ramirez's infield hit.
Addison Reed made the lead hold up with a 1-2-3 bottom of the 10th, notching his second save. Hector Santiago (1-1) earned the victory as the Chisox rebounded after losing both ends of a doubleheader in Cleveland Monday.
A.J. Pierzynski drove two runs for Chicago.
The Indians scored three runs in the eighth to forge a tie, highlighted by Carlos Santana's two-run single.