Martin Brodeur made 21 saves as the New Jersey 1, on Sunday.
"We have ourselves in a good position, but we have a lot of work to do," said Brodeur about the Devils, who currently sit in sixth place in the Eastern Conference. "It's going to get harder as we go, but the boys are playing with a lot of passion. They're blocking shots and making a second effort."
Max Pacioretty tallied the lone goal for Montreal, while Carey Price stopped 22-of-25 shots in the Canadiens' third loss in four games.
New Jersey held a 2-0 lead entering the third period, but an early interference penalty on the Devils' Dainius Zubrus allowed the Canadiens to pull within one. David Desharnais sent a pass from behind the net to Tomas Kaberle, who tipped it to Pacioretty and he finished it off with a quick shot past Brodeur at the 1:46 mark.
The Devils, though, regained their two-goal cushion with 11:33 left in the third period with Taormina's slap shot from the right point that sailed through a crowd and into the twine.
Montreal would not get any closer from there, as Brodeur made eight stops in the final period to hang on for the win.
"We can't let any shifts slide, but we did tonight and that was definitely the difference," Pacioretty said. "Maybe it's just nerves that our season could be over before the playoffs, but it's no excuse at all. In fact, it should give us more motivation to go out there and play 60 minutes hard as a team."
The Devils took advantage of a late offensive zone faceoff in the first period to notch the lone goal of the opening frame.
Adam Henrique won the faceoff back to Ilya Kovalchuk, who sent a wrister from the top of the left circle that was redirected through the short side by Parise with 1:57 remaining in the first.
Another late goal sent the Devils into the second intermission with a two-goal lead when Patrik Elias' wrister from the high slot deflected off the body of Clarkson as he was setting a screen in front and went past the glove side of Price with 17.4 seconds left in the period.
Brodeur made four saves in the first period and blocked nine shots in the second to keep the Canadiens off the board.
New Jersey recorded its eighth straight victory in Montreal. The Devils have not lost there since March 11, 2008...New Jersey took the season series by a mark of 3-1...Taormina's goal was his first of the season...Montreal went 1- for-3 on the power play, while the Devils did not have a power-play opportunity in the contest.