The Rangers were looking to advance to their second final in as many years and with an electric home backing at Madison Square Garden came into the decider as favourites.
But Ben Bishop made 22 saves and Alex Killorn and Ondrej Palat both netted in the third period to send Tampa to the showpiece series for the first time since 2004.
“I don’t think we were intimidated by the crowd tonight,” Bolts captain Steven Stamkos told www.nhl.com. “I think we embraced the challenge. We said that a lot, about going out and having fun.
“We had a little talk before the game about going out there and sticking to our structure no matter what happened.
“Give them some credit; their goaltender made some unbelievable saves early and kept this game 0-0 going into the third.
“You never know what’s going to happen. We stuck with it. That’s why I’m so proud of these guys.”
The Lightning will face either the Chicago Blackhawks or the Anaheim Ducks in the final.