Then, with one out in the eighth, Beckham belted the first pitch he saw from Mendoza deep into the left-field stands for the decisive blow.
Kevin Youkilis added a run-scoring double for an insurance run as the American League Central-leading White Sox took the opener of a three-game set and maintained a 1 1/2-game divisional lead over second-place Detroit.
Beckham's blast made a winner of Chisox starter Chris Sale (13-3), who struck out seven over eight innings. The left-hander yielded eight hits and didn't issue a walk.
Addison Reed tossed a 1-2-3 ninth for his 20th save.
Mendoza surrendered four runs on seven hits through 7 1/3 innings for the Royals.