Guthrie (4-0) notched his fourth victory without a loss over six starts to begin the season, issuing only one walk while striking out three on his way to the 106-pitch whitewash -- his 17th straight start without a loss going back to last season, breaking the franchise record set by Paul Splittorf in 1977-78.
The right-hander also continued his dominance against Chicago, yielding only two earned runs in 44 2/3 innings over his last six starts against the squad for a 0.40 ERA.
Lorenzo Cain provided the game's only offense with two-run triple in the first inning as the Royals took the opener of a rain-delayed three-game series.
The victory was the sixth in seven games for Kansas City.
White Sox starter Dylan Axelrod (0-2) absorbed the hard-luck loss, surrendering two runs on eight hits over 7 2/3 innings. The right-hander issued one walk.