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.
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