Kovalchuk took a pass from defenseman Marek Zidlicky low in the left circle and beat Michal Neuvirth, who was outstanding in overtime. Neuvirth made seven saves, most on a New Jersey power play.
Stephen Gionta and Patrik Elias scored for New Jersey, which is off to its first 3-0 start since the 2002-03 season, the last time the Devils won the Stanley Cup.
Mike Ribeiro and Mike Green scored for Washington in the final 7:15 of regulation to force overtime. The Capitals (0-3-1) became the last NHL team to earn a point this season.
Neuvirth finished with 32 saves, stopping five shots after Green was called for interference in the opening minute of overtime.
Ribeiro had Washington's only shot against Martin Brodeur in the extra session, but it was a good one in close. Brodeur, who was within 7:15 of his 121st career shutout, made the save with his shoulder. The puck just missed the upper corner of the net.
Washington was seemingly out of the game trailing 2-0 until the Devils took a couple of penalties in the final 10 minutes of regulation, The Capitals took advantage despite playing their second game in two nights.
Ribeiro cut the deficit in half with a slam-dunk goal from the edge of the crease with the Capitals enjoying a two-man advantage.
Green, who was stopped on a great chance from between the circles in the opening minute, tied it with 2:56 left in the third period. Zidlicky failed to get the puck out of his zone, and Green beat Brodeur trough a screen with a shot from the right point for his first goal of the season.
Brodeur appeared headed for his second straight shutout since April 2010, but it wasn't to be.
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Under new coach Adam Oates, the Capitals are off to their worst start since dropping six straight in the 1993-94 season. Washington recovered then and made the playoffs before losing to the eventual Stanley Cup champion New York Rangers in the second round.
The Devils didn't have a lot of chances against Neuvirth early, but made the most of the them.
Gionta gave the Devils the lead at 13:55 of the first period, just a second after an effective Capitals power play expired.
Defenseman Mark Fayne started the rush by poking the puck to Gionta at the blue line. He carried into center ice and made a cross-ice pass to Jacob Josefson. Before being sandwiched by defenseman Tomas Kundratek and Troy Brouwer, the Swede swept the puck to Gionta, who beat Neuvirth one-on-one for his second career regular-season goal.
Washington nearly tied the game on a power play with roughly eight minutes left in the second. Ovechkin, who was moved back to left wing from right wing before the game, was stopped on a blue-line blast. Brodeur then turned away a backhand rebound attempt by Joel Ward, who had the primary assist on Ribeiro's goal.
Elias stretched the Devils' lead to 2-0 with a minute left in the second period when New Jersey had a 5-on-3 advantage. A shot from the point by Zidlicky hit the skate of teammate David Clarkson in front, and Elias put the rebound into a wide-open net.
NOTES: Brodeur finished with 26 saves. ... The 3-0 start is the Devils' best since starting the 2002-03 season with the same record. They won the Stanley Cup that year.
Press; By TOM CANAVAN]
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