The Devils are one of three teams in the NHL this season to have lost at least 11 games in overtime and Dallas had a chance to add to that total after Tyler Seguin tied the contest with 52 seconds left in regulation.
Elias, however, scored just 12 seconds after Jamie Benn was sent to the penalty box for tripping. New Jersey ended a two-game skid.
New Jersey took a 2-0 lead on goals in the first period by Dainius Zubrus and Travis Zajac. Benn got one back for Dallas with 9:48 to go in the second period and Seguin tied it with his 24th goal of the season.
Cory Schneider had to make just 14 saves to get the win. Kari Lehtonen allowed three goals on 23 shots.