The Kings have taken a 2-0 lead in their first-round playoff series against the team that had the best record in the NHL's regular season.
Los Angeles will host Game 3 Sunday.
Brown scored shorthanded with 9 seconds left in the first period to give the Kings the lead and had another goal in the second period while Vancouver had the man advantage.
The second-period tally gave Los Angeles a 2-1 lead and Brown completed the hat trick 8:30 into the third with a power-play goal. Trevor Lewis made it 4-1 with 5:09 to play.
Jannik Hansen delivered the only goal Vancouver managed during the game's first 56 minutes, but his frustrations boiled over late in the contest when he was hit with a 4-minute roughing penalty and a game misconduct. Vancouver's Maxim Lapierre also received a game misconduct moments later.