Palat received a backhand pass from Nikita Kucherov and scored short side just inside the post from below the right face-off dot.
"I think we shot the puck twice, 'Kuch' got in the slot and made a great pass to me," Palat said. "I was just trying to shoot it on net and the puck went in."
Palat's game-winning goal was his 10th career in the Stanley Cup Playoffs, passing Tyler Johnson for the most in Lightning history.
"Palat, we've seen this for years. He is a great kid, he doesn't say anything," Tampa Bay coach Jon Cooper said. "All he does is give you everything he has on every shift. He's played up and down the lineup, but he's got the ability to play that blue-collar game with white-collar players, and that's a great trait to have."
RELATED: [Complete Rangers vs. Lightning series coverage]
New York leads the best-of-7 series 2-1. Game 4 will be here on Tuesday.
"They were the better team tonight, overall," Rangers coach Gerard Gallant said. "It went right down to the wire, and they had a chance to win late in the hockey game. Disappointing, but we'll move on and get ready for the next one."
Kucherov had a goal and two assists, and Steven Stamkos had a goal and an assist for Tampa Bay, which is the No. 3 seed from the Atlantic Division. Andrei Vasilevskiy made 28 saves.
Mika Zibanejad had a goal and an assist, and Artemi Panarin had two assists for New York, the No. 2 seed from the Metropolitan Division. Igor Shesterkin made 49 saves.
"We're going to have to go back and review the game, it just ended. But that being said, we played a pretty tough road game," Rangers forward Chris Kreider said. "We put ourselves in a pretty good spot going into the third (up 2-1). At the end of the day, what we'll take away from the game is the things we need to do better going into Game 4."