Galchenyuk ended the pulsating game when he cleaned up after a strong shot from Tomas Plekanec bounced off New York goalie Henrik Lundqvist and silenced the Madison Square Garden crowd.
The Rangers, who were looking to take a 3-0 stranglehold on the best-of-seven series, sent the game into overtime when Chris Kreider levelled the score at 2-2 with a shot off the skate of Montreal defenseman Alexei Emelin with 29 seconds left that produced roars that rocked the Garden.
Kreider's goal came after Daniel Briere had given Montreal the lead with a shot that caromed in off Rangers defenseman Ryan McDonagh with three minutes remaining in regulation.
Montreal's rookie goalie Dustin Tokarski, starting in place of injured regular Carey Price, stopped 35 of 37 shots to outduel Lundqvist who allowed three of 28 shots to find the net.
"Dustin Tokarski was phenomenal tonight," said Canadiens coach Michel Therrien. "Without Tokarskis performance, probably the result would have been different."
Game Four of the series will return to Madison Square Garden on Sunday.
The Rangers dominated the opening period, taking 14 shots to four by the Canadiens, with 24-year-old Tokarski hanging tough under the pressure to keep the game scoreless until left winger Carl Hagelin broke through for New York.
Hagelin smothered a shot by Montreal's P.K. Subban and raced up the left side on a breakaway before sliding it across ice to Martin St. Louis.
Tokarski tipped the shot by St. Louis but Hagelin swooped in and swiped it out of the air and into the net for a 1-0 lead with 5:51 left in the period for his fifth goal of the playoffs.
Montreal tied it 3:20 into the second period when Andrei Markov fired a shot through Lundqvist after receiving a pinpoint pass from Max Pacioretty.
Tokarski, appearing in his second playoff game after playing in 10 regular season games, said his team mates helped him through.
"I was just going out there to play my game," said Tokarski. "The other guys blocked a lot of shots and battled really hard to the end."
Both sides grumbled about the lucky bounces their opponents got.
"They got a bounce, that's the way it goes sometimes," said New York's St. Louis.
"You've got to bury some of your chances and we failed to do so but we generated a lot, that's some of the good things we did tonight. We've just got to find a way to put the puck in the net."
Therrien said it was critical that Montreal kept the Rangers from surging to a big lead in the opening period.
"We weathered the storm and I felt our team played with more confidence as the game went on," he said.
"They got a break at the end of the game (on Kreider's goal). Without Tokarski's performance, probably the result would have been different."
Rangers coach Alain Vigneault also paid Tokarski praise.
"We played a good game, had some really good looks. Give credit to their goaltender. He played well," Vigneault said.
"This series is intense. We battled really hard tonight, came out ready to play. It was a hard-fought game. Next one is going to be the same."
- Sports & Recreation
- Ice Hockey
- Dustin Tokarski
- Henrik Lundqvist
- Chris Kreider
- Madison Square Garden