The shootout was tied 1-1 when Brown used a forehand fake to get Ryan Miller out of position, and then put a backhander into the net for the decisive score.
The Kings got goals in regulation from Ryan Smyth, Brad Richardson and Sean O'Donnell and two assists from Drew Doughty. Quick was 26-of-29 in goal for LA, which won its league-leading 10th shootout contest but finished a seven-game homestand with just three wins against four losses.
Thomas Vanek, Clarke MacArthur and Craig Rivet scored for the Sabres. Ryan Miller went 26-of-29 in goal for Buffalo, which has lost five straight shootouts.