Carey Price was strong in net with 23 stops.
Montreal knotted the game at the 2:40 mark of the third period, when Tom Pyatt dished into the slot for Metropolit's redirection.
The Habs went ahead at 7:24, when Rask misplayed a chip-in from Travis Moen and Lapierre swept the rebound home.
Darche gave the Canadiens a two-goal cushion with 2 1/2 minutes left in regulation, with a quick return after Rask kicked aside a Metropolit backhander.
Pouliot hit the clincher into the empty net with 1:27 to play, seconds after Rask was summoned to the bench.
The Bruins picked up the game's first goal on the power play, as Sturm managed to pry the puck from underneath Price and slip it home with 7:49 to play in the first.
Boston came up empty on two second-period man advantages, but Rask made eight saves to keep the margin at one.
Montreal has won four of five meetings over Boston this season...Prior to the game, the Bruins announced that forward Patrice Bergeron is day-to-day with a groin strain suffered while playing for Canada during the Olympics.