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.
Copyright by STATS LLC and The Associated Press
Arena: STAPLES Center Location: Los Angeles, California
Derek Armstrong (power play) Assist: Kyle Calder, Brian Willsie