The 23-year-old left wing helped rally the Kings in regulation, then scored in overtime to give Los Angeles a 3-2 victory over the Chicago Blackhawks.
The game-winner past Nikolai Khabibulin was Brown's 18th goal, setting a new season-high.
"I just made a quick move, a power move to the net," Brown said of his game-winner. "I got in tight and tried to get it over him."
Brown and Kyle Calder connected in the third period to rally the Kings from a 2-0 deficit and force overtime.
"You have to make good shots to beat a goalie of Khabibulin's caliber," Kings coach Marc Crawford said. "He made a great shot to beat him."
The Kings ended an eight-game losing streak Saturday by scoring three times in the third period for a 3-1 win in Colorado.
"Our goal is to wear down other teams," said Brown, in his fourth year in the league. "The last two wins we came back in the third and wore on teams."
The Blackhawks played before a season-high United Center crowd of 21,715 -- their second straight sellout and largest since drawing 21,811 on Dec. 27, 2001.
"It feels real good," Calder said. "It was a tough there for eight games and it's not what you want to go through, but the team stuck together."
Rookies Jonathan Toews and Jacob Dowell scored in regulation for the Blackhawks, whose four-game winning streak ended. Toews leads all NHL rookies with 15 goals and is second only to teammate Patrick Kane in points by a first-year player.
Khabibulin made 33 saves. Los Angeles' Jason LaBarbera stopped 30 shots.
"What happened tonight is we just didn't have enough experience after taking the lead," Khabibulin said.
Added Chicago defenseman Duncan Keith: "It's tough when you have a 2-0 lead and end up losing. We have to learn from it and realize when we're up 2-0 to keep it simple and play it safe."
Skille, a right wing who was selected seventh overall in 2005, was recalled from Rockford of the AHL on Saturday and skated in his first NHL game. He narrowly missed scoring in the second period when his shot hit the crossbar.
Neither goaltender was heavily tested in the scoreless first period, although Khabibulin made a close-in pad save on Derek Armstrong in the final minute.
Toews opened the scoring at 2:16 of the second. From the left edge of the crease, he knocked in a rebound of Seabrook's shot from the top of the slot.
Dowell, playing in his first game after missing 12 with a broken foot, made it 2-0 at 2:10 of the third. From the right circle, he one-timed Martin Lapointe's cross-ice pass.
Brown replied just 33 seconds later to cut it to 2-1. He beat Khabibulin on the glove side with a shot from the left circle.
Calder's power-play goal at 7:29 tied it at 2. He fired in a loose puck after Tom Preissing's shot had deflected off traffic in front of the net.
- The Blackhawks played without two regular defensemen, Brent Sopel and James Wisniewski, who sustained injuries in Chicago's 5-2 win over Nashville on Wednesday. Sopel will miss one-to-two weeks with a broken finger and Wisniewski will be out two to four weeks with a sprained knee.
- The Blackhawks recalled D Danny Richmond, as well as Skille, from Rockford of the AHL. Both played Sunday.
- Kings LW Mike Cammalleri missed his third game with a rib injury. Los Angeles D Kevin Dallman sat out his third game with a broken foot.