The Rangers, seeded number one in the Eastern Conference but pushed all the way by the Senators, survived a furious onslaught from the visitors to advance to the Eastern Conference semi-finals for the first time in four seasons.
New York's next opponents are the Washington Capitals, who also advanced the hard way by beating last season's Stanley Cup champions Boston in seven games on Wednesday.
"We were everywhere, diving and blocking shots like we've been doing all year," said goaltender Henrik Lundqvist. "It was a great team effort and it's important to enjoy this tonight but then we have to regroup and get ready for the next one."
Marc Staal and Dan Girardi scored New York's goals in the second period but it was Lundqvist who once again provided the steel for the Rangers' backbone with a string of reflex stops.
The Swede, one of three netminders nominated for this season's Vezina Trophy, was forced to make 26 saves as the Rangers showed signs of fatigue against the relentless Ottawa offence.
Ottawa's goal came from Daniel Afredsson, who found a way past Lundqvist with a power-play goal that cut the margin to one but counted for nothing in the end.
"It's an empty feeling," Afredsson said. "Especially when the game goes down to the last few seconds like it did tonight.
"It's over. It feels weird and tough."
The Senators missed their opportunity to wrap up the series in game six in Ottawa and extended Canada's long and frustrating wait to get their hands back on the Stanley Cup.
The last Canadian team to lift the Cup was Montreal in 1993.
Ottawa and Vancouver both lost in the first round.
The Senators dominated a scoreless first period but could not find a way past Lundqvist, then the game burst into life in the second with three goals in seven minutes.
Rangers defenseman Staal broke the deadlock at 4:46 when he combined with Ryan Callahan and Derek Stepan to beat Ottawa goalie Craig Anderson, who had been as impenetrable as Lundqvist earlier in the series.
With the momentum back on their side, the Rangers buried the puck into the net for a second goal less than five minutes later from a Girardi slap-shot.
Alfredsson's goal, also in the second period, gave Ottawa some hope and briefly silenced the raucous crowd but the Rangers defense held firm against waves of attacks.
"It was play-off hockey at its finest," said Callahan.
"They are making a last surge, trying to play for their season and we stood in there and blocked some shots."