Archer (9-19) allowed three runs and six hits in 6 2/3 innings to help the Rays snap a six-game skid and leave Detroit's Mike Maroth as the only pitcher to lose 20 games this century.
Mikie Mahtook hit a solo homer in the fourth off Jose Quintana (13-12) and doubled and scored in a three-run seventh off Chicago's shaky bullpen.
Quintana allowed two runs and five hits over six innings for the White Sox, whose five-game winning streak ended despite Carlos Sanchez's two-out, two-run homer in the seventh.
Chicago clinched its fourth consecutive losing season under manager Robin Ventura, whose future with the team is uncertain.
