Moustakas supplied a run-scoring single in the third inning, staking the Royals to a 3-0 lead, and iced the game with a three-run homer in the eighth.
Butler, meanwhile, provided RBI singles in the third, fifth and eighth innings and Eric Hosmer added a solo shot for Kansas City, which cliched its three-game series.
Bruce Chen (9-10) earned the victory with six solid innings, during which he surrendered two runs on five hits while walking one and fanning five.
Adam Dunn cracked his major-league leading 35th homer -- a two-run shot -- in the eighth, cutting Chicago's deficit to 5-4, but the White Sox could get no closer.
Chisox starter Jake Peavy (9-9) allowed five runs (three earned) on nine hits over 5 1/3 innings.
Paul Konerko added a two-run blast for the American League Central-leading White Sox.
Astronomers offer more expansive view of universe
Duggar sisters unveil Christian dating rules in new book