Ryan Nugent-Hopkins, Andrew Ference, Perron, and Taylor Hall all scored their first of the season in a 7:47 span to make it 4-3. But despite getting a power play with 2:51 left in regulation, the Oilers (1-2) were forced to go to overtime.
Jason LaBarbera made 21 saves through the extra session and stopped both Devils attempts in the shootout. Brodeur made 23 saves, but was 0 for 2 in the tiebreaker.
"We dominated for most of the game and then took a tough penalty, and from there it got a bit chaotic. It was a tough one," Brodeur said. "Things like what happened don't happen too often.
"We are a veteran team and we should be able to play better under that kind of pressure. They are a good, young team, but we had them exactly where we wanted them."
The victory helped the Oilers avoid their worst start since 1995 when they were 0-4.
"It's tough to explain, but I'm glad we bounced back the way we did," Perron said. "To lose the first three would have been a pretty tough way to start the year. I know everyone was pretty eager to get that first win, and now we can look forward to new things and keep moving."
While the Devils were still a man down late, New Jersey pulled Brodeur for an extra skater, and Patrik Elias netted the tying short-handed goal with 54 seconds remaining.
Ference, the Oilers' 14th captain, said his team just needed to get over early-season jitters.
"We just want to win so bad. You see it with teams at the start of the year if you can't get in the win column right away," he said. "You can't be out there scared to lose. There is an element of fear in every person that plays sports, but that has to be outweighed by the desire to win.
"If you lose one game, it's not hanging over your head, but I felt that they were feeling the challenge of us having lost two games and they didn't want to get it to three," he said. "I just asked them to stay with it and believe in themselves, and they did.
"They finally got rewarded for their hard work and their patience."
Jaromir Jagr scored his first goal for the Devils (0-1-2), and Damien Brunner added his third of the young season to give New Jersey a two-goal lead in the first. Michael Ryder made it 3-0 in the second.
Nugent-Hopkins, making his season debut following shoulder surgery, started the Oilers' comeback 5:43 into the third after a mad scramble during a power play in front of the Devils net.
That spoiled Brodeur's shutout bid and left the Oilers as one of only three teams never to be blanked by Brodeur.
Edmonton made it 3-2 at 9:28 when Ference scored on a point shot through traffic.
The momentum shifted wildly, and the Oilers tied it at 11:47 when Perron plucked the puck out of a faceoff in the Devils end and picked the corner for his first goal with Edmonton since being acquired from St. Louis.
The Oilers took the lead just 1:43 later. Ales Hemsky chipped a puck off the boards to Hall, who was alone in front and slipped the puck between Brodeur's legs.
"It's a learning experience for us early in the season, and we've got to be better," Devils coach Peter DeBoer said. "I don't like leaving points on the table like that, especially after being up 3-0.
"The puck bounced right for them in the third period, and they capitalized, to their credit. It was a strange game."
But the Devils staged a rally, too, to rescue one standings point. Elias threw a puck on net that banked off Ference's skate and into the net to get the game to overtime.
Jagr put the Devils ahead on the game's first shot 2:59 in, and Brunner made it 2-0 at 8:01.
After a giveaway deep in the Oilers' end, the Devils had a 2-on-1 break. Ryder elected to take the shot and fired the puck into the top corner.
New Jersey won 2-1 at Edmonton in the last meeting between the teams on Jan. 11, 2012. ... Brunner has scored three of the Devils' first five goals this season. ... With New Jersey playing at Vancouver on Tuesday, Brodeur got the start against Edmonton. Former Canucks goalie Cory Schneider will start against his old club in his return to Vancouver. ... The Oilers scratched veteran forward Ryan Smyth, who is healthy, in favor of recent call-up Ryan Hamilton.
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