The Rangers killed all 11 penalties called against them, and Lundqvist made a game-saving stop on Robert Nilsson's shot with 2:19 left.
"He made some big saves, especially when he needed to," Islanders coach Steve Stirling said. "We had plenty of scoring chances, but the penalties early on were the difference."
The first-place Rangers (5-3-3) moved three points ahead of the third-place Islanders (5-5) in the Atlantic Division.
"We were lucky to win the game," Jagr said. "Bad penalties don't help anybody. A lot of guys end up killing penalties, getting tired. Guys can't get on the ice, and the rhythm is not there. It affects everybody."
The Islanders swept a home-and-home series from the Rangers last week, including a shootout victory at Madison Square Garden.
Jagr set up Martin Straka for his first goal of the season at 13:22 of the first period. With the Rangers on a power play, Jagr sent a gorgeous pass from behind the goal line to Straka, who beat helpless goalie Rick DiPietro from the slot.
"We weren't able to weather that first period without letting them score," DiPietro said. "They came out flying. It was a special teams game and they outplayed us on the [penalty kill]."
Fedor Tyutin scored his second goal of the season at 16:26 to give the Rangers a 2-0 lead. Tyutin carried the puck into the offensive zone and left a drop pass for Michael Nylander, who sent a cross-ice pass to Jagr. DiPietro stopped Jagr's slap shot, but Tyutin pounced on the rebound and scored.
Tom Poti appeared to give the Rangers a 3-0 lead midway through the second period, but a video review showed that the puck was blocked by the left leg of Islanders forward Trent Hunter before it crossed the goal line.
"It's tough to come from behind on the road, especially at Madison Square Garden," Campoli said. "We finished the game pretty strong but we ran out of time."
Blair Betts scored his second goal of the season into an empty net at 19:29 to complete the scoring.
"The last 10 minutes we got so pumped up. The game got faster and faster and I think everyone worked hard," Lundqvist said. "I think we went the extra mile to win the game, especially since it was so tight. To lose this game would have been bad."
Former Ozzy Osbourne guitarist Zakk Wylde played the national anthem as an electric guitar solo, a la Jimi Hendrix, before the game. ... Jagr has the second most points against the Islanders (112), behind only Mario Lemieux (128). Jagr has racked up 49 goals and 63 assists in 70 career games against the
Islanders. ... The Rangers have four former Islanders on their roster: Darius Kasparaitis (1992-93, 1996-97); Straka (1995-96); Jason Strudwick (1995-96, 1997-98); and Kevin Weekes (1999-2000). Mike York is the only former Rangers player on the Islanders (1999-2000), (2001-02).