The Kings won their second straight. Anaheim has lost four in a row and its nine wins for the season ties the Ducks with Columbus for fewest in the NHL.
Anaheim forced overtime when Lubomir Visnovsky scored with 3:11 to play in regulation.
Jarret Stoll scored for the Kings in the first round of the shootout, but Corey Perry immediately answered for the Ducks. After Brown scored in the third round, Jonathan Quick stopped Kyle Palmieri to secure the victory.
Brown also scored in the third period to give Los Angeles a 2-1 lead.
Quick made 20 saves. Jonas Hiller stopped 32 shots before giving up the two shootout goals.