New York erased two one-goal deficits to set up a 2-2 game before forward Chris Kreider slipped by Winnipeg defenseman Dustin Byfuglien on a rush and sent a backhand shot past goaltender Ondrej Pavelec with 3:46 left in the third period.
The Rangers (48-21-7), who clinched a playoff berth last Thursday, moved past the Montreal Canadiens into the Eastern Conference lead with 103 points. New York is eight points ahead of the Pittsburgh Penguins for first place in the Metropolitan Division.
Winnipeg (39-26-12) is two points ahead of the Los Angeles Kings for the second wild card into the playoffs from the Western Conference, but the Kings have a game in hand.
The Jets, who have lost two straight, are trying to reach the playoffs for the first time since 2007, when the franchise was known as the Atlanta Thrashers.
"Emotions are all high at this point," Jets coach Paul Maurice said when asked about the message he delivered to his players. "We just talked about how we're going to proceed."
New York's 25 road wins lead the NHL and match the franchise record, set last season. The Rangers have won seven of their past eight road games and are 22-6-1 in their past 29 games away from Madison Square Garden.
"We [had not] really been playing our brand of hockey," Kreider said. "[Before playing Winnipeg], we were rushing. We weren't patient, weren't poised, and I think we got back to the way we want to play tonight. It was more of a Ranger game."
After allowing four goals in a road loss against the Boston Bruins on Saturday, Lundqvist bounced back against Winnipeg. He made 15 saves in the first period and fended off the Jets' third-period push.
"I'm happy with how the guys played in front of [Lundqvist] and were able to get the late goal to get him the win tonight," Rangers coach Alain Vigneault said.
Pavelec made 21 saves.
"We didn't start the way we wanted," Vigneault said. "They had a really strong push. The good thing is that it got [Lundqvist] in the game early, I guess."
Lee Stempniak, who the Jets acquired from the Rangers in a trade March 1, put Winnipeg back on top 1:25 into the second period.
Derick Brassard's power-play goal at 10:13 of the second erased another one-goal lead for the Jets.
The Jets pounced on the Rangers early in the game. Winnipeg had a 10-3 shots advantage through the first seven minutes of the game, but Lundqvist stopped several in-close opportunities.
"This was a great test for us," Lundqvist said. "I thought that [Winnipeg] played a great game. They had speed. They played physically. They made it really tough for us, but I'm happy with the way we responded. After each [Winnipeg] goal, we settled in and started playing our game, our fast game."
Winnipeg took a 1-0 lead when Slater squeezed his bouncing rebound past Lundqvist. Two of Slater's four goals have come in the past three games.
"We just tried to weather the storm," said Rangers defenseman Dan Girardi, whose 10 blocked shots Tuesday were the most for the Rangers since the League started tracking the stat in the 2005-06 season. "We knew they would come out hard."
The Jets went ahead 2-1 when Stempniak took Mathieu Perreault's drop pass and snapped a shot under the crossbar for his 12th goal. Stempniak has three goals and three assists in 13 games with Winnipeg.
After Perreault was called for interference against Rangers center Dominic Moore, Brassard stepped into a left-side shot and sent it through Nash's screen for his 17th goal. Entering the game, the New York power play had gone 2-for-33 in its past 15 games.
Winnipeg does not play until Saturday, when a visit from the Vancouver Canucks finishes a four-game homestand.
"I think it's a good thing," Pavelec said of the schedule break. "After those two losses, mentally it's tough. Five games left, we have to prove everything we've got, and we'll see what happens."
"It's just one of those games that we're right on the edge all game," said Jets center Bryan Little, who returned after missing 11 games because of an upper-body injury. "You can't say we didn't have chances, or it might have been a different game.
"They did control the play sometimes, and I thought we did a good job of kind of riding the storm out. It honestly felt like they didn't get much, but the stuff we did give them, they took advantage of and scored on."
When the Jets threatened, Lundqvist stopped them. He is 26-12-3 with a 2.29 goals-against average and .921 save percentage. Having him ready for the playoffs is one of the Rangers' major goals for their final five games of the regular season.
Lundqvist did not like his play on Slater's goal, but his response encouraged him.
"I didn't change anything after that, and that's a good sign for me," Lundqvist said. "It's important that [I did not] change the game and stayed focused with the things I had to do, so that feels good."
After the loss at Boston, Lundqvist worked with goaltending coach Benoit Allaire on his movement and timing.
"I felt so much better tonight than the last game," Lundqvist said. "Just my focus and being more comfortable, it's night and day.
"I think that every game is going to help me to get closer to where I want to be mentally and technically. I feel a huge improvement from the [game at Boston], just the way I felt and my focus on the right details. It's definitely a step in the right direction for me."
The Rangers finish their two-game road trip when they visit the Wild on Thursday. A schedule down the stretch that includes games against teams battling to make the playoffs should help New York round its game into shape, Lundqvist said.
"It was a fun, intense game," he said. "It was a good test for me and the team to bounce back here. We needed this to feel good about our game."