Stamkos scored twice, and Ondrej Palat had two assists for Tampa Bay, the No. 3 seed in the Atlantic Division and two-time defending Stanley Cup champions. Andrei Vasilevskiy made 20 saves.
"When you are in, it's just sitting up here right now and thinking of the magnitude of going to a third Stanley Cup Final," Lightning coach Jon Cooper said. "You grew up as a kid, and I was not near good enough to play in this league. But growing up in Canada, you always dreamed about having your name on the Stanley Cup.
"And to get there the first time was, I mean, it was a dream come true. To get there a second time, like the next year, it was like a dream. Like there's no way we're going back, and to go a third time. Think about that. And you got to look at the players. And I sit back, and I'm just impressed. I'm impressed by them. It's impressive what they've done."
Frank Vatrano scored for the Rangers, the No. 2 seed in the Metropolitan Division. Igor Shesterkin made 29 saves.
"I don't know. I feel like denying this right now," New York forward Mika Zibanejad said. "I don't honestly have much to say. I'm just ... empty. Don't want it to be over."
New York scored five goals in the four straight losses after winning the first two games of the series by a combined 9-4. It had one even-strength goal in the final four games.
"I don't think we were expecting to score six goals a game after that first one," Rangers defenseman Adam Fox said. "Two great goalies. I don't think anyone expected this series to be a ton of goals. We knew special teams were going to play a factor. You want to score more 5-on-5, but we weren't able to do that. Obviously it hurt us a little bit."
[Complete Rangers vs. Lightning series coverage]
Stamkos put the Lightning ahead 2-1 at 13:28 of the third period off an odd-man rush. Shesterkin saved the initial shot but the rebound bounced off Stamkos' leg and into the net.
"Best captain in my life," Vasilevskiy said. "Just amazing."
Stamkos gave the Lightning a 1-0 lead at 10:43 of the second period. He took control of the puck in the neutral zone, cut toward the top of the circle and sent a sweeping shot that beat Shesterkin to the far post.
"I was just confident in our group that we would find a way, whether it was in regulation or was in overtime," Stamkos said. "We deserved to win the game tonight. And that was the feeling that we had from puck drop throughout the game. Just keep going, our will is going to take over."
The Rangers needed a dynamic performance from Shesterkin to keep the game close despite being outshot 31-21.
He made a diving stop on a chip from Nikita Kucherov at 1:20 of the second period and stopped Brandon Hagel at 6:29 of the third period to keep it a one-goal game.
"I thought Igor was outstanding," Rangers coach Gerard Gallant said. "He gave us a chance to win every night. We got out-chanced in a lot of the games, but he was outstanding. He was our best player all year long and again in this series."
