Quentin went 3-for-4, Paul Konerko added a two-run clout and Tyler Flowers contributed his first major league home run as the Chisox ended a seven-game home losing streak.
Jesse Crain (7-3) earned the win with 1 1/3 innings of scoreless relief and Chris Sale tossed a perfect ninth to rack up his fourth save of the season.
Kansas City's Luke Hochevar (8-9) was touched for five runs on eight hits over six-plus innings.
The Royals lost for the seventh time in nine games despite a two-run single from Billy Butler.
Swim Week Miami Beach 2014 [PHOTOS]