The Bruins, 1-9-3 in their last 13 games, blew a two-goal lead, then extended the second-longest losing streak in franchise history. They also lost their eighth straight home game, also the second-longest streak in club history.
Boston has scored 14 goals during the nine-game streak, 11 in the last eight.
The Canadiens got 45 saves from Jaroslav Halat in regulation and overtime and three more in the shootout. Gionta put Mongtreal's third bid past Rask, who made 23 saves.
Montreal beat the rival Bruins for the third time in three tries this season.
Mark Recchi and Blake Wheeler gave the Bruins a 2-0 lead, but Glen Metropolit and Roman Hamrlik scored 39 seconds apart late in the second period to tie it.
The two-goal lead was the first for the Bruins in eight games, since they took a 2-0 lead in Los Angeles on Jan. 16. They eventually lost that game in a shootout, starting the losing streak.
The Bruins, 1-for-6 on the power play, failed to score with a 4-on-3 advantage in overtime, and Rask had to stop Andrei Markov's short-handed bid during the power play.
Recchi tipped home Dennis Wideman's slapper at 15:48 of the first period. After the goal, Halak came up with several big saves, one on Michael Ryder, as the Bruins continued to dominate.
The Bruins made it 2-0 when David Krejci carried the puck down the right side and his centering pass hit Halak's pad and went right to Wheeler, who scored his first goal in eight games and 13th of the season.
The Canadiens, looking like a beaten team, were quiet before striking late in the period. Boston's Matt Hunwick was off for hooking when Metropolit, a former Bruin, slid home a rebound for his first goal in 19 games. Shortly after, Hamrlik let a wrist shot go from inside the left point and Rask, screened on the play, saw the puck fly over his glove for Hamrlik's first goal in his last 20 games.
Scott Gomez assisted on both goals.
The Bruins continued to have the better of the offensive play in the third period, including three by Marco Sturm, but also wasted two power plays. They had a 42-21 shots advantage heading into overtime and had five more shots in the extra 5 minutes.
NOTES: The game marked the 702nd regular-season game between the rivals, who meet again Sunday in Montreal. ... Krejci and Tomas Plekanec of the Canadiens will be teammates on the Czech Republic team and Bruins Zdeno Chara and Miroslav Satan will team up with Halak for Slovakia at the Olympics.
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