The Kings have won four straight and are 8-1-1 in their last 10 games. They moved out of a tie with Nashville and remained one point behind fourth-place Phoenix in the Western Conference.
"For the most part we gave up few chances," Brown said. "[Oilers rookie Jordan] Eberle had a good one in the third.
"Even with [Kopitar and Williams] in the lineup, our success has been in the defensive zone."
But even with a strong defense, goals will need to come from somewhere. Former Oilers forward Dustin Penner admits he feels pressure to contribute more than he has since his deadline-day trade.
"Everybody should have a bit more pressure on them," he said. "When you lose two guys like that, as a team you have to play better offensively. We are pretty sound defensively and in net but it looks like we will have to score by committee now."
The Oilers (23-42-11) have lost 10 straight, their worst skid since a 13-game slide last season.
"We didn't do a great job keeping the pace in the first period," Oilers coach Tom Renney said. "We needed to be better, and had we done that the outcome could have been a little bit different. We certainly gave it a good try at the end."
Edmonton's Nikolai Khabibulin was the busier of the two goalies during the scoreless first period. Khabibulin made 18 saves, the best of which was a partial breakaway stop on Kings captain Brown.
The Kings broke the deadlock five minutes into the second period when Trevor Lewis slid a puck under Oilers defenseman Ladislav Smid to Brown, who had an easy tap-in for his 26th goal.
Los Angeles used the same combination to go up 2-0 with five minutes remaining in the second. Lewis stripped Jason Strudwick of the puck and fed it in front to a streaking Brown for his second of the game.
Bernier had his biggest save eight minutes into the third period when he made a quick glove stop on a shot from the slot by Eberle.
Edmonton came close again with four minutes left when rookie Teemu Hartikainen rattled a puck off the post.
The Kings outshot Edmonton 33-32.
The Kings were without top scorer Anze Kopitar, who broke an ankle in Colorado on Saturday. ... Edmonton was without Shawn Horcoff, Sam Gagner, Ryan Whitney, Taylor Hall and Ales Hemsky. ... Former Oilers forward Dustin Penner made his first appearance against Edmonton since being dealt at the trade deadline. Penner had two goals and four assists in his first 12 games for Los Angeles.
Copyright by STATS LLC and The Associated Press
Arena: Rexall Place Location: Edmonton, Alberta
Referees: Gord Dwyer, Chris Lee Linesmen: Lonnie Cameron, Bryan Pancich
Attendance: 16,839 (100.0% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
No scoring this period
Dustin Brown (26)
Assists: Trevor Lewis, Ryan Smyth