Sale (1-0), making his first-ever Opening Day start, was tough in the cold weather at Chicago's U.S. Cellular Field, scattering seven hits while issuing only one walk and striking out seven.
Addison Reed threw a scoreless ninth to pick up the save for the White Sox, who lost the American League's Central Division crown to Detroit over the final two weeks of the 2012 season.
Flowers' solo homer in the fifth inning provided the game's only run.
James Shields (0-1) yielded one run on eight hits over six frames. The veteran right-hander didn't allow a walk and fanned six in his debut for the Royals.