Peavy (7-7) tossed 6 1/3 scoreless frames, walking two and striking out nine for the second-place White Sox, who remained eight games behind Detroit in the American League Central.
Rios connected for a solo homer in the fourth for a 2-0 Chicago lead, while Alejandro De Aza and Brent Morel added RBI for the Chisox.
Liam Hendricks (0-1) pitched well in his major league debut for the Twins, but took the loss, yielding three runs on four hits in a seven-inning effort.
Texas principal bans speaking Spanish, stirs controversy
18-year-old elf alleges mall Santa pinched her buttocks on the job