Dominic Moore scored the game's only goal in the second period as the Rangers wrapped up the Eastern Conference finals series 4-2.
The Rangers will play either the Los Angeles Kings or the Chicago Blackhawks for the National Hockey League's championship when the Stanley Cup finals start on June 4.
We've given ourselves the opportunity to compete for the Cup," said Rangers' coach Alain Vigneault.
"It was a great feeling tonight. We played, in my book, probably our best game of the play-offs."
New York goalie Henrik Lundqvist, who was surprisingly pulled during Tuesday's Game Five loss in Montreal, made a spectacular return with 18 saves and his first shutout of the play-offs.
He pulled off one fantastic save from Thomas Vanek in the second period when he dropped his stick but flipped around just in time to stop the puck going into the net, whipping the Big Apple's hockey fans into a frenzy.
"It was a roller coaster mentally," said Lundqvist, who gave up four early goals before he was taken off in the last game.
"But then you just have to make up your mind. I kept telling myself all day, 'believe in what you're doing' ... you have a tough game, but you just have to stay confident."
With Lundqvist back to his impenetrable best and the Rangers offense fuelled by a rowdy crowd at Manhattan's most famous old arena, the Blue shirts came out with all the ferocity of a prize fighter looking for a quick knockout.
Montreal's rookie goalie Dustin Tokarski soon found himself under fire, forced to save the first seven shots of the contest, including a vicious snap from Mats Zuccarelolo that rifled into his face mask.
But the Rangers failed to capitalise on their early chances - including a golden opportunity to score off a short-handed wraparound - and the Habs slowly began to grow in confidence in a scoreless opening period.
The Rangers came desperately close to scoring in the second period when a shot from Derek Stepan, playing with a broken jaw he suffered in Game Three, hit the post, triggering some anxious groans from the team's long suffering fans.
The Canadiens, who won the last of their 24 Stanley Cup championships a year before the Rangers' last success in 1994, killed off two New York penalties then almost snatched the lead when Lundqvist acrobatically stopped Vanek's backhander.
"When you get close to achieving a goal it hurts," said Montreal coach Michel Therrien. "It hurts more when you're close.
"But we made some good progress this year. I'm proud of this hockey team."
Moore finally broke the deadlock less than two minutes before the end of the middle period when Brian Boyle got in behind the Montreal net and fed the puck to Moore, who slipped it past Tokarski for his third goal of the play-offs.
The last time the Rangers made the Stanley Cup finals was in 1994 when they beat the Vancouver Canucks in seven games to end a 54-year title drought.
The odds seemed stacked against them this season when they needed seven games to win their first round play-off against the Philadelphia Flyers, then clawed their way off the canvas from 3-1 down to upset the Pittsburgh Penguins in the second round.
Within minutes of clinching the Eastern Conference, the Rangers' three co-captains, Brad Richards, Dan Girardi and Marc Staal, were presented with the Prince of Wales Trophy.
But in keeping with one of the sport's superstitions, none of the Rangers players dared touch the gleaming silver trophy, posing only for a team photograph before skating away empty handed, with their eyes on an even bigger prize.
"If you had told me in October that we were going the Stanley Cup finals, I would've said 'what are you smoking?'" joked Vigneault.
- Sports & Recreation
- Ice Hockey
- Henrik Lundqvist
- Dominic Moore
- National Hockey League