Montreal will play the New York Rangers for a spot in the Stanley Cup finals.
Dale Weise, Max Pacioretty and Daniel Briere scored for the Canadiens, who advanced to the conference finals for just the second time since winning the most recent of their NHL-record 24 Stanley Cup titles in 1993.
The Bruins won the Stanley Cup in 2011 and returned to the finals last year, losing to the Chicago Blackhawks in six games. Boston finished with the best record in the regular season this year, but its road stopped against the Original Six rival in another classic playoff series.
In front of boisterous crowds on both sides of the border, the Canadiens won the 34th playoff series between the two clubs - the ninth that went the seven-game distance. And, despite giving up home-ice advantage, Montreal quieted the Boston fans quickly.
Just 2:18 into the game, Briere dug a pass out of the left corner and centered to Weise, who redirected it past Rask. The Canadiens made it 2-0 midway through the second period when David Desharnais chased down a loose puck to set up a 2-on-1 with Pacioretty, who one-timed it past Rask.
Boston made it a one-goal game when Torey Krug's wrist show was deflected into the net by Iginla just 2:02 into the second period. But despite outshooting Montreal 21-12 over the final 40 minutes, the Bruins were not able to tie it up.
NOTES: The teams last played a Game 7 in the first round in 2011, when the Bruins won in overtime en route to the Stanley Cup title. ... The team that scored the first goal won every game.... Boston coach Claude Julien tied Mike Keenan with 10 Game 7s, the most in NHL history. ... Chara's 11 Game 7s are the most of any active player. ... Iginla's goal snapped Price's shutout streak of 103:46.