Brady Skjei tied it 3-3 for the Rangers at 5:31 of the second when his shot pinballed off defenseman Mike Matheson in front of the net and got past Montembeault.
Dadonov scored a power-play goal at 4:42 of the second to give the Panthers a 3-2 lead.
"As has happened throughout the course of this short season, when we get some goals, all of a sudden our mindset continues to change," Rangers coach David Quinn said. "We get very sloppy, we get very pond hockey-ish at times and if we trade chances with a team like that, rarely are we ever going to get a point."
Filip Chytil tied the game 2-2 for New York at 1:28 of the second when he skated around defenseman Anton Stralman and beat Montembeault.
Barkov tied the game 1-1 with 1:47 remaining in the first period when he took a pass from Dadonov in the right face-off circle, stopped, and beat Lundqvist with a wrist shot high to the far corner. Florida went ahead 30 seconds later when Dominic Toninato put a rebound past Lundqvist to make it 2-1 at 18:43. It was his first goal of the season.
"I thought we weathered the storm and played well in overtime and then got the win in the shootout," Vatrano said. "So it's obviously a good all-around effort for all of us."
Lindgren scored to give the Rangers a 1-0 lead at 5:06 of the first period, converting a cross-crease pass from Ryan Strome.