Alexei Ramirez had a three-run homer and four RBI and Dayan Viciedo added a solo shot for Chicago, which snapped a two-game losing streak.
Zach Putnam (2-0) earned the win in relief and Jake Petricka recorded the final two outs in the ninth to nail down his first career save.
Royals starter Jason Vargas (4-2) yielded seven runs on eight hits over 4 2/3 innings in the loss.
Eric Hosmer finished with three RBI and Lorenzo Cain knocked in two runs for the Royals, who had won two straight entering the contest.
Celebrity Families of 2014 [PHOTOS]