Kartye gave New York a 1-0 lead at 4:02 of the first period when he sent a snap shot past Halverson on the short side thanks to a pass from Zibanejad.
Kartye scored again at 1:29 of the second period to extend the Rangers' lead to 2-0. He got the puck off Miller's zone entry play that caromed off the end boards, and put a sharp angle snap shot over Halverson's stick from the left side.
"Ever since I got here, I feel like we've played pretty well," Kartye, who was claimed off waivers from Seattle on Feb. 27, said. "Even some games that we lost, I thought that we deserved a better result. But pretty much every game we were in, we were working hard, that's all you can ask for."
Gabe Perreault gave the Rangers a 3-0 lead at 4:49 with a wrist shot from the left point that got past Halverson on the glove side thanks to a screen in front.
Bjorkstrand got the Lightning on the board to make it 3-1 at 11:15, collecting a rebound from Scott Sabourin's original shot and tucking it home on the backhand as he fell to his knees in front.
"One of our goals was to make (the playoffs), and now we're here and it's back to zero," Lightning defenseman Darren Raddysh said. "Happy this regular season is over and I'm ready for the postseason."
Zibanejad added a power-play goal at 12:46 to push the New York lead to 4-1 on a tip from the slot off a touch pass by Alexis Lafreniere.
"I thought it was a really good game for the group," Garand said. "Sometimes when you're in this position, those are hard games to play, so I'm really proud of the way that we fought."