New York closed out a homestand with consecutive wins over the New York Islanders and the Lightning, outscoring the opponents 9-2. Nineteen-year-old rookie J.T. Miller scored two goals on Thursday, and second-year forward Hagelin matched him Sunday.
"That's what we needed in the beginning," Hagelin said of the secondary scoring. "Our first line was scoring all of our goals, and you're not going to win like that. It's important for every guy to chip in."
Lundqvist was sharp throughout and was stellar in making New York's early two-goal lead stand up.
"I really feel the most important part of the game was Hank in the first period," Tortorella said. "We're up 2-0 and we're giving up chance after chance. We could easily have been down 3-2."
Hagelin scored his first goal 31 seconds in and then made it 3-0 in the second period. Ryan Callahan added a breakaway goal with 4:47 left in the first, and Arron Asham and Nash scored in the third for the Rangers (6-5), who have won four of six overall and four of five at home.
The Lightning -- who entered with an NHL-best 42 goals -- have netted only six during their skid. Tampa Bay (6-5) might have been a bit weary following a long bus ride from Boston on Saturday after its game against the Bruins was postponed by the big Northeast snowstorm.
"This was unacceptable. Embarrassing," forward Steven Stamkos said. "We're a lot better team than this. If we want to be a good team, we have to play better than this. We had our chances in the first. After that it was downhill."
Vincent Lecavalier scored a second-period goal for the Lightning, who changed goalies after New York took a 3-0 lead on Hagelin's second goal at 9:58. Backup Mathieu Garon was pulled after stopping 16-of-19 shots and was replaced by Anders Lindback, who finished with 11 saves on 13 shots.
Asham pushed the lead to 4-1 by scoring his first goal with the Rangers at 12:22 of the third, and Nash netted his third of the season off a feed from Hagelin with 8.3 seconds left.
After netting at least three goals in their first seven games, the Lightning have failed to reach the mark during their losing streak that started with a 3-2 home defeat to the Rangers on Feb. 2. Tampa Bay is 1-4 on the road.
"We were awful in the third period ... trying to cheat and get back," Lightning coach Guy Boucher said. "We didn't deserve it. Guys are frustrated and they should be.
"This has been the same problem with us for four games. We can't capitalize on quality scoring chances, and they did."
Lecavalier ended Lundqvist's shutout bid with 5:18 left in the second period when he put in a rebound of defenseman Victor Hedman's shot from the left point. That concluded a wild sequence in the Rangers' end that began with a 4-on-1 rush by the Lightning after a bad line change by the Rangers.
Lundqvist stopped a few drives before Lecavalier finally beat him for his fifth goal of the season. Alex Killorn, recalled Sunday from Syracuse of the AHL, made his NHL debut and earned his first point with an assist.
Despite six odd-man rushes by the Lightning in the first period, including a failed breakaway by Lecavalier, the Rangers held a 15-6 shots advantage.
"They had a couple of good chances in the first, but after that we settled things down and really played a really strong game," said Lundqvist, who has played in 10 of the Rangers' 11 games. "We have a lot of confidence, and we're really relaxed and starting to play our game."
Lundqvist was tested early. Lecavalier had the best chance just more than 3 minutes in when he came in alone. As he cut from right to left, the puck slipped off his stick and went wide.
Tampa Bay then had a 3-on-1 rush at 11:06, again with Lecavalier leading the way. This time, he fed a pass to Cory Conacher, who lifted a close shot into Lundqvist's midsection.
The missed chances came back to haunt the Lightning, who contributed directly to the Rangers' second goal.
Matt Carle's clearing attempt went right to Callahan in the neutral zone. He calmly came in alone on Garon and scored his third of the season and first in two games since returning from a shoulder injury that forced him to miss three games.
Rangers D Dan Girardi returned to the lineup after missing two games because of an undisclosed injury that ended his consecutive-game streak at 145, dating to Feb. 1, 2011. Steve Eminger sat out to make room for Girardi. Veteran forward Brian Boyle was scratched for the second straight game. ... Lightning LW Ryan Malone was a late scratch because of a lower-body injury. He is day to day. Pierre-Cedric Labrie was in the lineup. ... The Lightning assigned forward Dana Tyrell to Syracuse. ... Tampa Bay hasn't scored a power-play goal in three games after connecting eight times on 16 chances. The Lightning were 0 for 2.
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NEW YORK, NY - FEBRUARY 10: Steven Stamkos #91 of the Tampa Bay Lightning practices during warm-ups prior to the...