Parise was left alone behind Montreal's defense to take Travis Zajac's long pass at the Canadiens' blue line before driving in to put a backhander past Jaroslav Halak 2:38 into the extra period.
Zajac scored in the first, and Martin Brodeur made 29 saves for his NHL-record 585th career win. The Montreal native is 37-15-5 in 57 starts against the Canadiens.
Former Devils forward Scott Gomez scored a power-play goal in the second for the Canadiens, who have lost three of four.
It was the first time the Devils have played beyond regulation since Dec. 5. Each of New Jersey's six previous non-regulation games this season were decided by shootout.
Halak stopped 26 shots, including a spectacular pad save on Zajac in the third to keep the game tied. Halak recorded his sixth career shutout — his second of the season — in Thursday's 2-0 win over Florida.
Before the game, the Devils learned they will have to play for the third time in three nights on Sunday when their suspended game against Tampa Bay will be completed. The game was halted in the second period on Friday, with the Devils trailing 3-0, because lights in half the arena turned off and couldn't be turned on again.
Brodeur played at the Bell Centre for the first time since March 14, when he tied Patrick Roy's NHL record of 551 regular-season wins in a 3-1 victory over the Canadiens.
New Jersey opened the scoring this time 4:07 in on Zajac's second goal in two completed games. Zajac, who scored in New Jersey's 4-0 win over Dallas on Tuesday, put a backhander through Halak's pads for his 11th goal.
Gomez tied it at 1 during a power play 6:28 into the second period. He slapped home a puck that was loose in the crease after Roman Hamrlik's wrist shot from the point caromed off Brodeur's blocker and hit the right post.
Montreal recalled right winger Ryan White from Hamilton of the AHL on Saturday because Georges Laraque was unavailable because of an upper-body injury. Andrei Kostitsyn, who had already missed three games, had arthroscopic surgery on his left knee Friday and will be sidelined for six weeks.
Devils D Mark Fraser had a spirited fight with Montreal's Travis Moen 12:58 in after he was checked into the boards. … The Devils began what was originally scheduled as a five-game trip. After completing the suspended game on Sunday, New Jersey will play at the New York Rangers on Tuesday. … Canadiens LW Sergei Kostitsyn was sidelined for a fifth game by a lower-body injury. D Ryan O'Byrne missed his fourth straight because of family matters.