Karns (1-2), who was acquired in a Jan. 6 trade from Seattle for Jarrod Dyson, picked up his first victory since June 25.
He walked one and struck out seven, throwing 62 of his 95 pitches for strikes to lower his ERA to 4.97.
Karns struck out four White Sox in the fifth. With two out, Yolmer Sanchez fanned on a wild pitch, but reached first. He stole second base and Cabrera walked. Jose Abreu went down swinging to end the inning.
Karns became the first big league pitcher to strike out four in an inning this season. The last Royal pitcher to accomplish that was Kevin Appier, Sept. 3, 1996, at Toronto.
After Karns exited, Peter Moylan threw a spotless eighth. Matt Strahm gave up the homer to Abreu in the ninth.
White Sox right-hander Mike Pelfrey (0-2) matched Karns with zeroes on the scoreboard until the sixth when the Royals scored three times.
Whit Merrifield's single with one out started it. He scored on Mike Moustakas' double to right-center. Lorenzo Cain's run-producing triple to right-center ended Pelfrey's night.
He was replaced by left-hander Dan Jennings, who gave up a sacrifice fly to Eric Hosmer that center fielder Leury Garcia tracked down at the center fielder wall.
Pelfrey's final line was three runs on five hits and a walk over 5 1/3 innings with one walk and one strikeout.
Jennings yielded two additional runs in the seventh with Alcides Escobar's double down the left-field line scoring Jorge Bonifacio. Merrifield's sacrifice fly scored Brandon Moss with the final run of the inning.
Alex Gordon's two-out eighth inning double off Chris Beck plated Hosmer for the last run.
Chicago (14-12) dropped out of first place in the bunched up American League Central. Kansas City (9-17) has won two out of three in May after ending April on a nine-game slide.
NOTES: Royals OF Jorge Soler, who is on the disabled list with an oblique strain, went 2-for-5 with a home run in a rehab game Wednesday for Triple-A Omaha. He is hitting .343 with three consecutive multi-hit games in 10 rehab games. ... The White Sox have scored 57 runs over the past 10 games. ... White Sox LHP Derek Holland and Royals RHP Ian Kennedy are the Thursday probables for the four-game series finale.