Brown scored the Kings' first three goals and helped out on the fourth from Willie Mitchell as Los Angeles ended a four-game losing streak.
Jonathan Quick tied the New York Rangers' Henrik Lundqvist for the NHL lead in shutouts with seven, turning aside all 19 shots he saw.
The streaky Blackhawks, meanwhile, dropped their second straight following four consecutive wins.
Corey Crawford started the game and allowed four goals on 10 shots before being pulled in the middle frame.
Ray Emery made 20-of-20 saves the rest of the way.