Bryan Smolinski and Tom Kostopoulos also scored for Montreal, which finished first in the Eastern Conference with 104 points, in large part thanks to their season sweep of Boston.
"We couldn't have asked for a better start than the one we got, two quick goals," Montreal left wing Chris Higgins said. "The fans were already into it but they got way more into it after that."
Carey Price also made his playoff debut, stopping 17 shots for the Canadiens, who won all eight regular-season meetings between the two longtime rivals.
"It was really loud out there, as usual," the 20-year-old rookie said. "You could hardly hear yourself think and you almost don't have enough time to be nervous."
Bruins 33-year-old goalie Tim Thomas stopped 28 shots in his first playoff game.
"I don't think we played well, that was the bottom line," Boston coach Claude Julien said. "Their physicality was a testament to how badly they wanted to win the game."
Montreal will host Game 2 of the best-of-seven series Saturday night.
Shane Hnidy scored in the first to make it 2-1 after Sergei Kostitsyn opened the scoring 34 seconds in and Andrei Kostitsyn gave the Canadiens a two-goal lead at 2:02.
Smolinski, who debuted with Boston in the 1992-93 season, scored 5:26 into the second to restore Montreal's two-goal lead at 3-1.
It was Smolinski's 23rd playoff goal in 112 playoff games, and his first with the Canadiens.
Kostopoulos scored his first playoff goal 7:24 into the third.
Montreal captain Saku Koivu did not play because of a broken bone in his left foot that sidelined him for the final four games of the season.
The East champions' blue line got a boost from the return of Mike Komisarek. The team's leader in hits and blocked shots, Komisarek missed the last seven games of the season because of a hip injury.
"It felt good," Komisarek said. "I felt better as the game went on."
The Canadiens got off to a fast start, taking a 2-0 lead to the delight of the sold-out Bell Centre crowd of 21,273.
Sergei Kostitsyn beat Thomas off an unfortunate rebound for the Bruins' goalie. Montreal defenseman Patrice Brisebois' shot from the right point struck Boston defenseman Aaron Ward's leg and Thomas made a pad save on the redirection but was unable to stop the younger Kostitsyn's shot off the rebound.
Kostitsyn's older brother, Andre, also got his first playoff goal quickly in his postseason debut. Linemate Tomas Plekanec set up the goal with a fine centering pass from the right side to Andre Kostitsyn, who beat Thomas with a shot from the slot.
The Canadiens came close to making it a three-goal lead on their first power play opportunity moments later, though it was Boston that struck next despite being outshot 8-1 in the early going.
|Sergei Kostitsyn (1) ASST: Patrice Brisebois (1)|
1 - 0 MTL
|Andrei Kostitsyn (1) ASST: Tomas Plekanec (1)|
2 - 0 MTL
|Shane Hnidy (1) ASST: Andrew Ference (1), Phil Kessel (1)|
2 - 1 MTL
|Bryan Smolinski (1) ASST: Tom Kostopoulos (1), Mike Komisarek (1)|
3 - 1 MTL
|Tom Kostopoulos (1) ASST: Maxim Lapierre (1), Mark Streit (1)|
4 - 1 MTL
|Shane Hnidy Interference against Sergei Kostitsyn|
|Andrei Markov Delaying Game - Puck over glass|
|Glen Murray Holding the stick against Mike Komisarek|
|Mike Komisarek Hooking against Glen Murray|
|Guillaume Latendresse Roughing against Jeremy Reich|
|Jeremy Reich Roughing against Guillaume Latendresse|
|Michael Ryder Interference against Marc Savard|
|Marc Savard Hooking against Chris Higgins|
|Shawn Thornton Cross checking against Steve Begin|
|Mike Komisarek Interference against Milan Lucic|
|Zdeno Chara Hi-sticking against Tomas Plekanec|
|Marco Sturm Slashing against Tomas Plekanec|
|Milan Lucic Cross checking against Mike Komisarek|