Darren Helm had the lone tally for Detroit, which dropped the final three games of a five-game road trip that started in Buffalo and ended on Saturday in Phoenix. Detroit lost to Anaheim, San Jose and Phoenix to end the trip.
Jonathan Quick was spectacular in stopping 28 shots for the Kings. Quick was pulled after giving up six goals on 18 shots in a 7-4 loss last week to Detroit.
Helm gave Detroit a 1-0 lead 5:57 into the game, sliding a soft shot through Quick.
Los Angeles tied it just under eight minutes into the second period when Kopitar knocked home a loose rebound past Red Wings goalie Jimmy Howard. Brown scored the winner in somewhat controversial fashion, tipping a shot from the point past Howard for the lead on the power play.
Moments earlier, Red Wings captain Nicklas Lidstrom had been high sticked in the face by a Kings player, but no penalty was called. The incident cut Lidstrom and drew blood, which is an automatic four minute penalty if caught by officials.