The Kings came into the finals after steamrolling Vancouver, St. Louis and Phoenix in just 14 games, and they made the Devils look ordinary in the first 40 minutes, holding them to nine shots.
But a fluke goal by Volchenkov tied the game with 1:12 left in the second.
Volchenkov took a shot from the left point that Quick kicked away in front. The puck went airborne, avoided Elias in front and hit off the shoulder of Kings defenseman Slava Voynov before going into the net.
Until then, Fraser's first career playoff goal was beginning to look like the winner.
It was a typical Kings' goal, created off the forecheck — by the fourth line, no less.
Jordan Nolan checked New Jersey's Andy Greene behind the Devils' net, outfought him for the puck and found Fraser between the circles for a shot that beat Brodeur 9:56 into the game.
The Kings had chances to extend the lead, but Brodeur, who was the difference in the Devils' victory over the rival Rangers in the conference finals, made three good saves. The best stop by the 40-year-old, three-time Cup winner came on the opening shift of the second, when he blocked Kopitar point blank on the edge of the crease.
Kopitar, of course, would get one past Brodeur much later in the evening.
Brodeur also stopped forward Jeff Carter from in close and made a big pad stop on Penner in the second.
The Devils were held without a shot for more than 14 minutes of the period before Parise was credited with one on a short-handed attempt in which the puck rolled off his stick into the crease.
Quick, who wasn't very busy in the first two periods, made his best save with a glove stop on Zubrus from the left circle after a turnover.
However, New Jersey managed to tie it on Volchenkov's strange goal.
The tally came just after Quick got into a tussle with Parise in the crease, and refused to let the Devils' captain get up after he fell as Parise lost his helmet.
"I think we're going to have to find another level," Devils coach Peter DeBoer said. "We were a little tentative, as was expected for a Stanley Cup final. I think we got better as the game went along, but you have to play 60 minutes against that team.
"We have to be better than we were tonight."
NOTES: Less than a minute into the game, the chant of "BEAT L-A" echoed through the arena. ... Doug O'Neill, the trainer of Kentucky Derby and Preakness winner I'll Have Another, was on hand, as was New York Giants offensive lineman Chris Snee and New York Jets coach Rex Ryan. ... Volchenkov's goal was his first in the playoffs since Game 3 of the 2007 Stanley Cup finals when he played for Ottawa against Anaheim. ... Kopitar's goal was his second OT winner in the postseason. The other was in 2010 at Vancouver. He has 12 points in the last nine games, including six goals. ...The was the first Game 1 of the finals that went to overtime since 2002, when the Carolina Hurricanes defeated the Detroit Red Wings 3-2 at Joe Louis Arena on a goal by Ron Francis, just 58 seconds past regulation.