After scoring in each of his last two games, Elias tied John MacLean's franchise record with his 347th goal on a power play 5:05 into the second. He broke the record with his second tally of the game -- and 13th of the season -- to give New Jersey its third one-goal lead 1:31 into the third.
"It just feels nice," said Elias, who was MacLean's teammate at the outset of his career. "It just feels that you're doing something right and it keeps me going. Hopefully, I have a lot more in me, obviously, but at this moment I'm enjoying it."
Petr Sykora and Dainius Zubrus, who gave New Jersey a two-goal lead with a power-play goal midway through the third, both assisted on the record-breaking goal in addition to scoring themselves.
"It's good that it happened on the game-winning goal in Montreal, and we kind of made a play as a line," said Sykora, Elias' longtime teammate who returned to the Devils this season. "Zubie, great on the forecheck, and then we kind of 2-on-0 the goalie. The whole play and the whole meaning of that goal, I'm very happy that I'm a part of it, just to help him with the helper."
Cunneyworth took over behind the Montreal bench as the team's interim coach after Jacques Martin was fired earlier in the day.
"I would have enjoyed it more if we won," Cunneyworth said.
David Clarkson also scored for the Devils, who have won six of seven, including a 6-3 win over Dallas on Friday.
Martin Brodeur, whose father was the Canadiens' team photographer, made 23 saves to improve to 41-18-5 against Montreal in his career.
"Offensively, the power play, I think we did a lot of good stuff again," Brodeur said. "Defensively, it's still a battle. (Friday) night we had a lot of quality chances and again tonight, but I think we have to expect that a little bit for the way we play. It's been good. We've got to keep it going."
"We wanted to start fresh but it's the same story over again," Eller said. "But there are still a lot of games left and we have to pull ourselves together and figure out a way to win."
Clarkson scored with 1:34 left in the second to tie it at 3 after the Devils blew 1-0 and 2-1 leads earlier in the period.
Sykora gave New Jersey a 1-0 lead 7:05 in, just after the Canadiens had killed a tripping penalty to Alexei Yemelin.
The Canadiens tallied three times in the middle period to take a 3-2 lead after erasing a pair of one-goal deficits.
Subban one-timed a slap shot from the left point through traffic and past Brodeur to tie it 2:21 into the second.
Elias' first goal of the game restored New Jersey's one-goal margin before Campoli evened it at 2 with his first goal in four games with Montreal at 6:33.
Eller scored his third goal at 8:57 to give the Canadiens' their only lead.
Canadiens right wing Erik Cole was called for three straight minor penalties late in the first and early in the second, including an illegal check to the head on Devils rookie Adam Larsson with 1:07 left in the opening period.
Larsson remained in the game and assisted on Clarkson's goal.
"Those are the hits you're trying to take out of the game," Devils coach Peter DeBoer said. "We're fortunate that we got out without him being hurt."
Elias, who played his 993rd game, all with New Jersey, is the Devils' all-time points leader with 845. MacLean is second with 701 points. ... Brodeur, the NHL's all-time wins leader with 633, played his 1,150th game. ... It was the 15th time Gill has had a career-high two points in 1,022 regular season games.