Brown's wrist shot late in the first period broke a 1-1 tie and the Kings' captain added an empty-netter to seal the victory, which completed a 4-0 sweep of the Blues.
With the win, the Kings reached the conference finals for the first time since 1993. There they will meet the winner of the series between Phoenix and Nashville.
Jonathan Quick turned aside 23-of-24 shots, improving to 8-1 in the post-season.
Kevin Shattenkirk scored the only goal for the second-seeded Blues.
Brian Elliott made 16 stops in the loss.