Rios, who had entered the game in the first inning after right fielder Carlos Quentin hurt his left shoulder while making a diving catch, chased home pinch-runner Brent Lillibridge with a single to left off Darren Oliver in the eighth.
Koji Uehara (1-3) was saddled with the loss after allowing Paul Konerko's one-out single.
Alexei Ramirez and Juan Pierre each had an RBI as the White Sox snapped a three-game losing skid and took the middle contest of a series with the Rangers.
John Danks started for Chicago and pitched well, allowing two runs on seven hits over seven innings, but Jesse Crain (8-3) got the victory with a scoreless eighth.
Ian Kinsler led Texas at the plate, going 3-for-4 with two solo homers as American League West-leading Rangers lost for the second time in three games.