With the game scoreless in the top of the 10th, Billy Butler drew a two-out walk against Brett Myers (2-3) and Salvador Perez, Moustakas and Francoeur each delivered singles to give the Royals a 2-0 lead.
Kelvin Herrera (3-2) earned the victory as the Royals took the rubber match of a three-game series.
The American League Central-leading White Sox maintained a two-game cushion over Detroit following the Tigers' loss Sunday to Los Angeles.
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