On the final attempt of the shootout, Brown ripped a shot past James Reimer and off the post into the net.
However, it is probably the final game for the team under Stevens as Darryl Sutter will likely be introduced as the new head coach on either Tuesday or Wednesday.
Sutter agreed to become the next head coach of the team on Saturday, but is still working on immigration issues. Stevens has been guiding the club on an interim basis after the team fired Terry Murray last week.
Los Angeles' next game is on Thursday at home against Anaheim.
John-Michael Liles and Matt Frattin each scored for the Maple Leafs, who have dropped three straight. Reimer stopped 40 shots.