Ondrej Palat, Tyler Johnson and Mikhail Sergachev scored, and Andrei Vasilevskiy made 31 saves for the Lightning, the No. 2 seed, who had won six straight games since losing 3-2 to the Boston Bruins in Game 1 of the second round on Aug. 23.
"It's a tough one," Tampa Bay defenseman Victor Hedman said. "We've got to put this behind us, learn from it and move forward. That's all we can do. We can't change what happened a few minutes ago."
Game 4 is in Edmonton, the hub city for the conference finals and the Stanley Cup Final, on Sunday (3 p.m. ET; NBC, CBC, SN, TVAS).
Palat redirected a centering pass from Kucherov on a power play to get the Lightning within 3-2 at 2:32 of the third, and Johnson tied it 3-3 by redirecting a shot from Erik Cernak at 12:04. Johnson's goal was upheld after a video review determined his stick was not above the crossbar.
Cal Clutterbuck gave New York a 1-0 lead at 12:58 of the first period on a wrist shot from the left face-off circle past Vasilevskiy, whose right pad had come loose earlier in the play.
Sergachev scored on a backhand in the slot to tie it 1-1 at 16:31.
"Obviously, whenever you play a playoff game, you don't like your opponent," Sergachev said. "You've got to be hard on him. You've got to take the puck away from them and score goals. We're doing that by hitting them, playing physical, playing fast. I don't know if it's a rivalry, but they don't like us, we don't like them."