Elias' seventh goal this season snapped a 1-1 tie at 17:24 of the third period, following assists by Brian Rolston and Vladimir Zharkov.
Ilkka Pikkarainen scored his first goal, while Martin Brodeur equaled boyhood idol Patrick Roy with his 1,029th regular-season appearances by an NHL goalie. Brodeur made 17 saves on 18 shots for the Devils, who have won six of seven.
Montreal's goal was scored by Travis Moen 7 (shorthanded, unassisted) at 12:23 of the first period.
Carey Price was beaten twice on 27 shots for the Canadiens, who have lost four in a row -- all by one goal.
On the power play, New Jersey went 0-for-5, while Montreal went 0-for-1.