Guthrie (1-3), who was acquired from Colorado in a July 20 trade, allowed five hits and struck out six in his fourth start for the Royals.
Guthrie had not won a start since May 31 but he set down nine batters in a row before Gordon Beckham doubled with two outs in the sixth.
Mike Moustakas and Salvador Perez backed Guthrie up with home runs and Greg Holland recorded his third save despite giving up an RBI single to A.J. Pierzynski in the bottom of the ninth.
Pierzynski and Alejandro De Aza both had two hits for the White Sox, who lead the American League Central by a half game over the Detroit Tigers.
Celebrity Families of 2014 [PHOTOS]