Hochevar dominated the White Sox, yielding only three hits and a walk while striking out five as he snapped a personal two-game losing streak.
The Royals right-hander had been pounded for 19 hits and 16 runs in his previous 6 1/3 innings.
Greg Holland and Aaron Crow completed the three-hit shutout with one scoreless frame each as Kansas City reversed Friday's 5-0 loss to Chicago.
Chisox starter Chris Sale (3-2) yielded three runs on seven hits over five full innings.
The White Sox saw their three-game winning streak end.
Celebrity Breakups and divorces of 2014 [PHOTOS]