Boston leads the best-of-seven series 3-2 after a second straight win, leaving Montreal one loss away from elimination from the Stanley Cup playoffs.
"I thought we've been battling really good here and we were down 2-1 (in the series) and now we're up 3-2, so, definitely a much better feeling," Boston right winger Loui Eriksson said.
"But still, we know it's going to be a tough game, the next game (Monday night in Montreal), so we have to be ready and focus for that."
Carl Soderberg, Eriksson's Swedish countryman and center on Boston's third line, scored his first NHL playoff goal and added two assists as the Bruins, with the help of two power-play goals in 32 seconds, built a 3-0 lead and never let Montreal get closer than two the rest of the way.
The power-play goals, which turned a 1-0 lead into a 3-0 advantage early in the second period, ended a run of 39 straight failed Boston playoff power plays against Montreal dating back to 2009.
Right wingers Reilly Smith and Jarome Iginla scored the goals, both on penalties against Montreal center Tomas Plekanec.
Right winger Brendan Gallagher and defenseman P.K. Subban scored their fourth respective goals of the playoffs for the Canadiens.
Soderberg has five points in the series, but this was clearly his best game.
"He's a big body out there, skating real well and making some good plays," Eriksson said.
Left winger Matt Fraser, the third member of the line, scored the overtime winner in Game Four, his first career playoff game, and had an assist on Saturday.
"You just kind of roll with it and it's amazing what a little confidence can do with a line, and they're fun guys to play with," Fraser said of the Swedes.
"They work well with the puck, and I just try and kind of read off them and find the open areas."
Soderberg scored his first career playoff goal when his shot from the slot hit Canadiens goaltender Carey Price and went in at 13:20 of the first period.
"It was just one of those games where we were doing something wrong on every side of the puck all night," said Montreal captain Brian Gionta.
(Editing by Andrew Both and Patrick Johnston)