Chris Kreider gave the Rangers a 1-0 lead 1:11 into the first period when he scored from the left circle off a pass across the middle from Zibanejad.
Stamkos tied it 1-1 at 7:18 with a slap shot from the top of the slot.
"The Rangers have some dynamic players, and you give them an inch and they'll take a mile, and they did that tonight," Lightning coach Jon Cooper said. "Give credit to them. They're a good team. ... They didn't fluke their way to where they are in this playoffs."
Vatrano put New York ahead 2-1 at 7:50, picking up a loose puck and sending a wrist shot over Vasilevskiy's right shoulder. Palat tied it 2-2 42 seconds later at 8:32 on the rebound of a shot by Stamkos.
"You've got to give them credit," Stamkos said. "That's a really good hockey team over there. They played well tonight. We certainly didn't have our best, so not much we can do about it now. ... Too many turnovers and execution wasn't there tonight. We have to respond. We've been in this position before and I'm confident our group is going to have a much better effort next game."
RELATED: [Complete Rangers vs. Lightning series coverage]
Panarin gave the Rangers a 5-2 lead 30 seconds into the third period, and Zibanejad scored a power-play goal for the 6-2 final at 6:06.
"It's always fun to play with the lead and not trying to crawl back into games," Vatrano said, "but they are a team that can score in bunches also, so we know no lead is safe against them. We've still got to do the little things right regardless of the score and can't give them any freebies. I think in the second and third period, we did a great job at moving our feet and being smart with pucks."
The Lightning allowed six goals after giving up three in a four-game sweep of the Florida Panthers in the second round.
Tampa Bay is 17-0 following postseason losses since the start of the 2020 playoffs.
"We've got better in us. We're going to show up in two nights from now," Cooper said. "… I give the Rangers full marks on their game, their plan, their effort tonight, and we'll try and make a few adjustments here and we'll be back in two nights."
NOTES: Vasilevskiy allowed six goals for the third time in his 93 playoff games. … The Lightning allowed two third-period goals after not giving up any in their previous six games. … The Rangers scored a power-play goal for the sixth straight game, matching their playoff record. They did it twice before (1986, 1990). … Jan Rutta had an assist on Stamkos' goal for his first point in these playoffs. It was the defenseman's sixth game.