Jake DeBrusk made it 2-0 at 13:38, eluding Montreal defensemen Joel Edmundson and Justin Barron to skate in alone on Allen and beat him with a low shot.
"It's one of those things where I'm just trying to build my game, as we all are, going into the playoffs," DeBrusk said.
Bruins forward A.J. Greer was assessed a five-minute major and game misconduct at 17:08 for cross-checking Canadiens forward Mike Hoffman in the face. Suzuki then cut it to 2-1 on the power play with 19 seconds left with a slap shot from the left face-off circle.
"Guys are sticking up for each other," Suzuki said. "It's a rivalry that goes back many years, and you saw that tonight right from the get-go. It was important for us to bring that energy against them in their arena, and I thought we did a good job of that."
The teams combined for 31 penalty minutes, 23 by the Bruins, in the first period.
"Physicality is part of the game," Canadiens coach Martin St. Louis said. "It's not like we're running around to go get physical, but there's a situation on the ice that presents itself that you have to be physical. Some games there's more of an opportunity than others. Early on, you saw a little bit of that, but I thought we stayed calm and collected and just kept playing."
After Swayman made a blocker save on Rafael Harvey-Pinard at 2:09 of the second period to help kill off the rest of Greer's major, Pastrnak pushed it to 3-1 27 seconds later at 2:36. He scored five-hole from the left circle after Bertuzzi intercepted Allen's clearing attempt along the boards.
Dach cut it to 3-2 with a power-play goal at 16:13, redirecting Matheson's pass on the doorstep.
"The way [the Bruins] kind of kill [penalties] is guys kind of go out in front," Dach said. "So, we were able to find that same backdoor and just angle my stick and hope 'Mikey' (Matheson) makes [the play]."