Scott Clemmensen, who made 25 saves, became the first New Jersey goalie other than Martin Brodeur - who had left elbow surgery in early November - to start against the Canadiens since Mike Dunham in a 4-4 tie in Montreal on Feb. 1, 1997.
Clemmensen came up with a fine save on Andrei Kostitsyn's shot from the right side with just over 2 minutes remaining in the third.
The Devils, who won 3-2 in overtime Thursday in Philadelphia, ended a five-game road trip with a 4-1 record. New Jersey has won seven of eight.
Montreal, which fell to 3-1 on its team-record seven-game homestand, lost two defensemen in the third, including Andrei Markov, who was struck in the mouth by a puck that deflected off Parise's stick at 12:08.
Markov left the ice hunched over and bleeding from the mouth and went directly to the dressing room, joining Mathieu Dandenault, who left the game moments earlier after he was struck on the right foot while blocking a shot.
Dandenault returned to the bench before overtime. Markov did not return.
Canadiens rookie Matt D'Agostini scored for the third game in a row to give Montreal the lead 15:35 in.
Clemmensen was unable to hold Markov's shot from the left point in his glove, and D'Agostini swatted the rebound into the net after the puck dropped at the New Jersey goalie's feet.
Saku Koivu recorded his 613th point with an assist on D'Agostini's goal to move ahead of Mats Naslund as the highest-scoring non-Canadian in team history, and into 11th place overall among the storied franchise's career leaders.
Jamie Langenbrunner got New Jersey even at 1 with his fourth goal in the second. The Devils captain drove a slap shot between Carey Price's legs that squeezed through the Montreal goalie's pads and dribbled behind him and rolled over the goal line 4:05 into the middle period.
Price came up with a huge save later in the period, stopping Patrik Elias' shot from the slot at 15:24.
