Escobar's first blast, a two-run shot in the third, gave the Royals a 3-1 lead.
Then, with the score knotted 3-3 in the seventh, White Sox starter Jake Peavy (7-6) retired the first two batters before Escobar lined the first pitch he saw into the left-field bullpen, putting Kansas City ahead to stay.
Greg Holland (3-2) earned the victory with 1 1/3 scoreless frames of relief and Jonathan Broxton closed out the victory with a 1-2-3 ninth for his 22nd save.
Peavy ended up allowing six runs on 12 hits over seven full frames.
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