Brian Rolston set up all three goals by Elias, and rookie Nick Palmieri got the go-ahead goal midway through the third period in what was a must-win game for the Devils.
New Jersey has to win its final five games and get a lot of help or it will miss the playoffs for the first time since 1996.
James Van Riemsdyk and Jeff Carter scored for Eastern Conference-leading Philadelphia.
In Columbus, Ohio, defending Stanley Cup champion Chicago secured a vital two points with a shootout win over Columbus.
Viktor Stalberg had the only goal in the shootout and Corey Crawford made 31 saves for the Blackhawks, who are eighth in the Western Conference, the last position for a playoff spot.
Patrick Kane had a goal and an assist and Brent Seabrook and Bryan Bickell also scored for Chicago.
Derick Brassard, Maksim Mayorov and Samuel Pahlsson scored for the Blue Jackets.
In St. Louis, Jarome Iginla scored his second goal of the game with 5:03 left to reach 1,000 points and help Calgary rally to beat St. Louis.
Iginla became the 77th player in NHL history to record 1,000 points when he scored to complete the Flames' comeback from a 2-0 deficit.
Iginla also had an assist. He has 39 goals and 41 assists in 79 games this season and 480 goals and 520 assists in 1,103 career regular-season games — all for the Flames.
Alex Tanguay added a goal and two assists for Calgary.
David Backes had a goal and an assist, and Matt D'Agostini also scored for the Blues, who were eliminated from playoff contention when Chicago beat Columbus.
In Glendale, Arizona, Ryan O'Reilly tied it late in regulation, then had the only score in the shootout to give lowly Colorado a victory over Phoenix.
Shane Doan had a power-play goal and assist as the Coyotes rallied from a two-goal deficit to take a 3-2 lead, but O'Reilly scored on Colorado's only shot of the third period to send the game into overtime.
Philippe Dupuis and Jay McClement also scored for the Avalanche, whose only three victories in the past 17 games have all come in shootouts.
Lee Stempniak and Ray Whitney also scored for Phoenix.
