With Kopitar, their leading scorer, lost for the season because of ankle surgery and Williams, second in team scoring, gone with a dislocated shoulder, the Kings made it four straight wins on the strength of Brown's two goals and 32 saves by goaltender Jonathan Bernier.
"You're not going to replace either one of those players," Brown said. "They're two of our best players. At the end of the day, hockey is a team game and we played a pretty good team game tonight."
The Oilers, last in the overall standings, lost their 10th in a row.
The Kings lost Kopitar and his 73 points in a 4-1 win over Colorado last Saturday. Williams had 57 points when he was injured in a 2-1 win over Calgary March 21. Costly victories, to be sure, but with performances like Brown provided against the Oilers, the Kings haven't buckled -- they remained one point behind fourth-place Phoenix, a 2-1 shootout winner against Dallas.
"We're going to have to play a real smart game, a checking game," Los Angeles coach Terry Murray said.
"I said this to the team when we came off the ice -- I think it answered a lot of questions to ourselves as far as what style, who we are and what we have to do. Guys really don't have step up and do anything extraordinary. They just have to play the game the right way and things will fall in place."
Los Angeles certainly could have used Kopitar and Williams in the first period against the Oilers. Aided by three power plays, they fired 18 shots at Nikolai Khabibulin -- but had nothing to show for them in a scoreless 20 minutes.
"We go what we expected, to be honest with you," Edmonton coach Tom Renney said. "We thought they would come out hard and put a real big push on to start the game.
"They did that, and I think, in part at least, because of what they don't have. Better to play the game in the other team's end than your own if you don't have some key people. We can kind of relate to that."
But Brown made it 1-0 at 5:11 of the second when he tapped a goalmouth feed from Trevor Lewis into a gaping net for his 26th goal of the season.
Brown and Lewis connected again on the second goal. Lewis stole the puck from Edmonton defenseman Jason Strudwick and found Brown, who buried a one-timer behind Khabibulin at 14:56.
"I think everyone shouldered a little bit of the load," Lewis said. "When you get more minutes and play a little bit more, you get more chances. I just have to keep going."
Two goals was plenty for Bernier, who did his best work with saves on rookies Jordan Eberle and Teemu Hartikainen in the third period on the way to his third shutout of the season and the fourth of his career.
"The guys really stepped up and we played a solid road game," Bernier said. "We'll have to do the same thing until the end.
"When a guy like Kopi gets hurt, I think guys want to take their chance and do well. I think that's what going to happen with this team, I'm sure."
The win gives the Kings a sweep of the four-game season series against the Oilers. It's the first time since Edmonton joined the NHL for the 1979-80 season the Kings have swept a season series from the Oilers.
"When we're at our best as a team, it's because of structure, not individual performances," Brown said.
"That's a positive for us. Like I said, you're not going to replace those guys and it definitely hurts losing them, but we've got to battle on as a team."
1 - 0 LAK
2 - 0 LAK