Martin Straka added two power-play goals for the Rangers, who improved to 6-2 on the road this season. Blair Betts had the other New York goal. Henrik Lundqvist, returning in net for the Rangers after being replaced by Kevin Weekes the last three games, made 37 saves before the shootout.
Ruslan Salei, Nathan Horton and Jay Bouwmeester scored for the Panthers, who trailed 2-0 midway through the second period.
But the Panthers were stymied in the shootout by Lundqvist. He stopped Olli Jokinen, Horton and clinched the victory when he blocked Joe Nieuwendyk's backhand attempt on a deke.
Nylander scored on the Rangers' first attempt in the shootout when he deked to his backhand and lifted the puck past Ed Belfour.
Belfour, who also beat the Rangers on Oct. 25, stopped 27 shots for the Panthers in regulation and overtime.
Betts gave the Rangers a 3-2 lead at 1:34 of the third on a spectacular individual effort. He came in on the wing and skated around defenceman Mike Van Ryn before his backhand flip was stopped by Belfour. The rebound came right back to Betts, who put the puck past Belfour.
Bouwmeester tied it at 9:57 when he skated in from the point and poked home a loose puck through a crowd in front of the net.
The Rangers had a 4-on-3 power play in overtime, but it was negated when Michal Rozsival was called for interference just 23 seconds after the Florida penalty.
Straka opened the scoring at 3:13 of the first when he one-timed a slap shot from the high slot and beat Belfour to the stick side.
Straka scored again at 2:07 of the second when his shot from the point went through a screen and again beat Belfour to the stick side.
Salei began the comeback at 11:13 of the second period on a four-on-four when he took a centring pass from Nieuwendyk and deked Lundqvist to his backhand.
The Rangers had a chance to add to their 2-1 lead before Horton's goal, but Petr Prucha's shot from close range went off the crossbar.
Horton tied it at 17:18 on a power-play goal on a second rebound after Van Ryn's slap shot from the points.
Notes: C Prucha, who was hit in the face by a puck in the Rangers previous game on Sunday, played with a full-length visor. ... Nieuwendyk returned to the lineup after missing the last four games because of a back injury. ... Salei's goal was his first in a Panthers uniform. He spent eight seasons with Anaheim before signing with the Panthers as a free agent in the offseason.