Dunn led off the second with a blast to right-center field and Tyler Flowers, Dayan Viciedo and Alexi Ramirez added home runs to account for all the White Sox scoring.
Jake Peavey (1-0) allowed four hits and two runs (one earned) in six innings in collecting the win and Addison Reed pitched the ninth for his second save for Chicago.
The first three Chicago home runs came off Kansas City starter Ervin Santana (0-1), who only gave up two other hits and struck out eight over six innings.
Dunn connected on the first pitch of the second inning and, after the Royals tied the score with the help of an error, Flowers hit the first pitch of the bottom of the third for his second home run of the year.
Viciedo's fourth-inning home run came after Dunn had walked and made the score 4-1. Kansas City scored on a force out in the sixth before Ramirez led off the seventh with his homer.