Budaj also stopped a four-game Bruins winning streak with a 4-1 victory in Boston on Jan. 30.
"We are very confident with him back there," Montreal forward Brian Gionta said. "That first period he made some huge saves and even in the second he made some big saves where they could have been back in the game real quick."
Boston's last loss came March 1, 4-2 to Washington.
Montreal grabbed the lead on Alexei Emelin's third goal of the season at 6:39 of the first. Emelin's shot from the blue line deflected off Bruins forward Chris Kelly about 30 feet from the net and rose over the glove of goalie Tuukka Rask.
Then Galchenyuk went in close to put the puck between Rask's stick and the post to the right of the goaltender.
"I was excited and all the moves were playing in my head," Galchenyuk said. "They were on a roll and we are happy that we're the team that ended that roll."
The Bruins had three power plays in the first 11 minutes of the third period but were outshot 5-1 as the Canadiens picked off errant passes and cleared the puck from their zone.
"Hopefully, next time when we get that many (power-play) opportunities we'll get more than one (goal)," Bruins forward Milan Lucic said.
The Canadiens won despite losing two forwards, Dale Weise and Travis Moen, early in the first period. Bruins defenseman Kevan Miller sent Weise sprawling into the boards 4:50 into the game then fought with Moen, who challenged him after the hit.
"I felt bad about it. I just kind of bumped into (Weise)," Miller said. "I stood there just trying to make sure he was OK for a second and Moen asked if I wanted to (fight)."
Montreal had an excellent opportunity with a two-man advantage for 44 seconds late in the second period when Johnny Boychuk was called for roughing P.K. Subban and joined Iginla in the penalty box. The Canadiens took four shots on goal before Iginla's penalty ended and none during the rest of the power play.
Budaj came up with a big glove save when he stopped Zdeno Chara's slap shot from the blue line with 5 seconds left in the second period.
Montreal C Lars Eller missed the game with a lower-body injury. He has played in 71 of his team's 73 games. ... Former Bruins defenseman Ray Bourque dropped the ceremonial first puck. ... Rask is 3-11-2 in his career against Montreal. Budaj is 5-2 in his career against Boston. ... The Canadiens are 31-0-3 when leading after the second period.
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BOSTON, MA - MARCH 24: Zdeno Chara #33 and alumni legend Ray Bourque of the Boston Bruins at the ceremonial puck...