Kopitar faked a backhand shot, put the puck on his forehand and beat a prone Martin Brodeur.
Los Angeles has won all nine of its road games in the playoffs, an NHL record. The Kings are now one win shy of tying the NHL record for postseason road victories.
More importantly, they are three wins away from the franchise's first NHL title. They have won 11 consecutive road playoff games dating back to last season.
Kopitar saw Justin Williams battling with Devils defenseman Bryce Salvador and forward Dainius Zubrus along the boards.
''I wanted to make sure I went to the middle,'' Kopitar said. ''I don't know if he heard me or not, but I yelled for the puck and he chipped it. It was perfect, right on my tape. It happened pretty quick and I was able to finish it off.''
As soon as he rifled the puck into the net, Kopitar raised his hands and banged himself into the boards, facing the crowd off to Brodeur's right.
''To put it past a goaltender like Marty is a good feeling,'' Kopitar said.
The veteran goaltender dejectedly skated off to the locker room as the rest of the Kings piled on Kopitar.
This the third straight series in which the Devils have lost the first game.
''I think it was probably the worst game in the playoffs for us,'' Devils leading scorer Ilya Kovalchuk, who was limited to one shot. ''Maybe we were a little too nervous before the game started, but it's all excuses. We've got to make sure we know what we didn't do right and be a different team next game.''
Fourth-line center Colin Fraser scored in the first period for the Kings, the No. 8 seed in the Western Conference who beat the top three teams to get to their first Stanley Cup Finals since 1993.
Anton Volchenkov tied it late in the second period for New Jersey, the East's sixth seed.
Kings goaltender Jonathan Quick finished with 17 saves in what was a relatively easy night. Brodeur had 23 saves as the Devils lost in overtime for just the second time this postseason; they have won four times. LA is 3-0 in overtime this spring.
The Devils had two great chances to take the lead early in the third period, and for a split second it appeared they went ahead with 16:02 to play in regulation when Zach Parisescored off a wild goal-mouth scramble.
While the horns went off and the fans celebrated, referee Dan O'Halloran quickly waved off the goal.
It was reviewed in Toronto and replay clearly showed Parise swept the puck into the net with his hand.
Defenseman Mark Fayne was probably kicking himself six minutes later when he missed a wide-open net from the edge of the crease in what was the Devils' best period of the night.
The Kings had their chances, too, with Brodeur making two outstanding saves about 10 seconds apart. He made a stacked-pad save on a one-timer by defenseman Drew Doughty from 30 feet after a drop pass from Mike Richards. A turnover seconds later set up Dustin Penner for a shot from the left circle.
The Kings came into the finals after steamrolling the top three seeds in the Western Conference 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 Devils forward Patrik 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.