It was the Devils' third win over Philadelphia in three games in the lockout-shortened season.
Ilya Kovalchuk scored 4:39 into the extra session, and the Devils won their third straight game to start the lockout-shortened season, 3-2 over the winless Washington Capitals on Friday night.
Fayne tied it at 2 at 5:46 of the third period while the Devils were killing a penalty.
The Red Devils have already come through a third-round match-up at Manchester City, after a 3-2 win at the Etihad Stadium.
Ilya Kovalchuk injured his right shoulder when he slammed into the boards behind the Florida Panthers' net in the third period of the Devils' 2-1 victory Saturday night.
The senior scored a career-high 36 points in his return from a foot injury that had sidelined him since January, helping the third-ranked Blue Devils beat the Hurricanes 79-76 on Saturday night in a matchup of the Atlantic Coast Conference's top teams.
With less than 12 minutes remaining in the third period and the Devils on a power play, Michael Grabner, the Islanders forward who is one of the fastest skaters in the NHL, picked off Ilya Kovalchuk's pass and cruised in alone on Hedberg.
" The second one is "Devils on the Doorstep, " and the third one is "The Sun Also Rises.
Despite outshooting the Senators 23-9 in the opening 40 minutes and getting an early two-man advantage for the second straight game, the Devils trailed 2-0 heading into the third period.
The win was the second straight and third in four games for the Devils following a six-game losing streak that dropped them close to the eighth-place cutoff for the playoffs.
Trailing for only the third time coming into the third period this season, the Devils came out storming with Ward stopping David Clarkson right on his doorstep in the opening seconds.
The Devils had a great chance to expand the lead in the third period when Florida defenseman Colby Robak was given a major penalty for cross-checking Tim Sestito.
But the Devils couldn't hold the lead and lost past regulation for the third straight game.
Craig Anderson made 33 saves in his third shutout of the season, and the Senators extended the Devils' losing streak to within one of the franchise record with a 2-0 victory Friday night.
With Carolina on the power play and the score tied at 2 in the third, Tlusty flipped a shot toward the net that bounced off Devils defenseman Anton Volchenkov and over goalie Martin Brodeur at 7:25.
After Reimer made huge saves on Adam Henrique and Patrik Elias that bookended Devils defenseman Andy Greene's drive that hit the post midway through the third, Kessel gave Toronto a 1-0 lead with a shot that beat goalie Martin Brodeur shortside at 13:28.
Tlusty scored two goals in the third period and Cam Ward made 26 saves as the Hurricanes beat the Devils 4-2 on Tuesday night, snapping New Jersey's five-game winning streak.
Henrique put the Devils up 2-1 with a short-handed goal 1:21 into the third period.
Madison Square Garden fans serenaded the Devils with chants of "Season's over" as time ticked down in the third.
Ryan Callahan scored a goal in the opening minute and another in the third period, and Henrik Lundqvist made 26 saves as the Rangers eliminated the Devils from the Eastern Conference playoff race.
The Devils also have to move on, but they need to change direction after losing their third straight (0-2-1) and for the 10th time (3-8-2) since goalie Martin Brodeur went down with a pinched nerve in his back.
James Reimer made 31 saves, and Phil Kessel snapped a tie late in the third period to lift the Maple Leafs to a 2-0 victory over the Devils on Monday night, despite recording only 12 shots.
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