Ethan Moreau gave Edmonton a 2-1 lead 3:35 into the third period with his second goal in as many games.
But Elias picked off a clearing attempt by defenseman Eric Brewer and wristed a shot from the blue line that Brylin tipped by goaltender Tommy Salo with 4:38 left in regulation.
In the extra period, Scott Gomez avoided a hit in the neutral zone and another inside the Oilers' blue line before pulling up at the outer edge of the right faceoff circle. Elias burst down the slot and deflected Gomez's centering pass over Salo's right arm for his team-leading 12th goal and third game-winner.
Turner Stevenson had the other goal for New Jersey.
Raffi Torres also scored for the Oilers.