After falling behind to Chris Neil's first-period opener, Eastern Conference top seeds New York exploded for three goals in the second to tie their series at 3-3 and avoid the same fate as West top seeds Vancouver, who were eliminated on Sunday.
Derek Stepan, Brad Richards and Chris Kreider scored unanswered second-period goals to overturn the 1-0 deficit and spoil the return of Ottawa's Daniel Alfredsson, who missed three games with a concussion.
The Rangers will host Game Seven on Thursday.
"“(Tonight) is the way we need to play and that's the way we're going to continue to play," Stepan said.
The Senators charged forward in the third period and got a controversial goal with 39 seconds remaining when Jason Spezza appeared to kick the puck into the Rangers net.
New York goaltender Henrik Lundqvist was left fuming that the goal was not chalked off.
"It's an absolute joke," said Lundqvist, who finished with 25 saves. "It scares me when it's such an obvious goal interference and a kick and they still call a goal."
In the Western Conference, Phoenix beat Chicago 4-0 to wrap up their series 4-2 on the back of netminder Mike Smith's 39-save shutout performance.
It was the first game of the series not to go to overtime and gave the franchise their first series win since moving to Phoenix in 1996.
"It's a huge achievement for our team. We've grown a lot," Smith said. "We're finding ways to win games Coyotes style."
Oliver Ekman-Larsson put the visitors ahead in the second period and the Coyotes out the game out of sight with three third-period scores from Gilbert Brule, Antoine Vermette and Kyle Chipchura.
It was Phoenix's third victory of the series at Chicago's United Center.
The Blackhawks had their chances on Monday, outshooting the Coyotes 39-20 but could not find a way past Smith in his first postseason shutout.