The win was the 17th career shutout for Hedberg, who faced only four shots in the third period and survived four Dallas power plays.
Nick Palmieri scored the game's only goal on a power play at 14:23 of the third period with Ilya Kovalchuk and Mark Fayne getting the assists.
Palmieri's goal spoiled an otherwise solid performance by Kari Lehtonen, who made 22 saves in the loss. Dallas fell to 1-8-1 for the month.
New Jersey has won eight consecutive games and 10 of their last 11. The win moved the Devils to within nine points of the eighth playoff slot in the Eastern Conference.