With the score tied 2-2 early in the final period, the Kings' Slava Voynov got control of the puck at center ice and fired a long pass onto the stick of Brown streaking into the Coyotes' zone.
The Los Angeles captain moved into the right circle and fired a quick wrist shot that got under the arm of Phoenix goalie Mike Smith for the decisive tally -- Brown's seventh goal of the post-season.
Dwight King scored in the second period and added an empty-netter while Anze Kopitar also tallied for Los Angeles, which improved to 6-0 on the road in the playoffs.
Jonathan Quick turned aside 25-of-27 shots to earn the victory.
Derek Morris and Mikkel Boedker scored for the Coyotes.
Smith finished with 44 saves.