The Oilers came into the final period trailing 3-0 but proceeded to reel off four unanswered goals to take the lead.
Perron assisted on Ryan Nugent-Hopkins' goal 5:43 into the frame and Andrew Ference found the net less than 4 minutes later. Perron then notched the equalizer with 8:13 left and Taylor Hall quickly followed with a go-ahead marker.
But the Devils' Patrik Elias forged a 4-4 tie with 54 seconds remaining, sending the game into a scoreless overtime and then to a shootout.
Jason LaBarbara stopped 21-of-24 shots to earn the victory.
Damien Brunner notched goal and an assist while Jaromir Jagr and Mark Ryder each scored for New Jersey, which remained winless on the season.
Brodeur finished with 23 saves.