The Kings had taken a two-goal lead early in the third period when Brown and Brady Murray each scored within a 3:12 span. Graham Mink quickly drew San Jose within a goal, scoring just 19 seconds after the goal by Murray.
Defenseman Doug Murray then tied the game with just 3:15 remaining in the third period.
Dan Cloutier stopped all three San Jose attempts in the shootout.
Torrey Mitchell scored twice for the Sharks and Devin Setoguchi added a goal and two assists.
The Kings will play in Colorado on Wednesday and Las Vegas on Saturday before traveling to Salzburg, Austria for a pair of games next week. They will open the regular season in London skating against the Anaheim Ducks on September 29 and 30.