After one, shots on goal were 18-11 for Montreal.
Jesper Kotkaniemi received a pass in close quarters, turned and rifled it from the right face-off circle to tie the game up 1:40 into the middle frame but once again Devils responded quickly.
Less than three minutes later, Miles Wood received a long feed from Pavel Zacha in the neutral zone and used his speed to break in all alone. He shot five-hole past Price to put the Devils back in front and 3:54 after that, Joel Armia tied the game for a third time for Montreal.
After Price made a nice save on a redirection in front by Hall, the play went back the other way where Armia rifled a shot from the left circle past Blackwood for the goal.
The fast goals kept coming. 2:26 after Montreal's tying goal, Jesper Boqvist scored his second of the season and his career, pouncing on a Nico Hischier rebound in the slot and firing it past Price to give the Devils their fourth lead of the night.
After two, shots on goal were 35-22 for Montreal.
Damon Severson gave the Devils a cushion just over eight minutes into the third. He skated into the Canadiens zone and rifled home a shot from the right circle to put the visitors up by a pair.
Sixteen seconds later, Artturi Lehkonen stormed the Devils net and put the puck home to make it a one-goal game again at 5-4.
Coleman added an empty-net goal to ice the outcome.
"They came hard all night. It felt like a bit of a track meet at times. Mackenzie was strong. He's been great for us and it was no difference tonight," said Coleman. "When you get those big saves, you build confidence for the team. He deserves a ton of credit for the win."
Devils head coach John Hynes said another big reason for the win was the team's ability to make plays in the face of a difficult Montreal defense.
"We made some good plays .They were coming hard. We were fortunate to make some plays to beat the pressure but there were other times that they had pretty good sustained pressure. But it was well-played with good passes when we had those opportunities."