Vesey gave New York a 3-1 lead at 15:03 of the second period by diving in from the right side to poke at the loose puck and having it go off Aaron Ekblad's stick into the net.
"It's always good when the puck is going in with not many shots on net, so of course it's better for our mood, excited, energy coming," Panarin said. "That's always nice when you shoot a few and score a few. That happens sometimes."
The Rangers opened up a four-goal lead in a matter of 14 seconds in the third period.
Lafreniere found the loose puck in the crease to make it 4-1 at 7:56 of the third period. He hadn't scored since Dec. 7.
"I think we had a pretty good shift and I just saw the puck by the goal line, just pushed it in," Lafreniere said. "Hopefully it gets me going a little bit."
Zibanejad scored off the rush at 8:10 with a shot from right circle to make it 5-1.
Barkov cut it to 5-2 at 11:02 with a shot that rolled down Shesterkin's back and into the net.
The Panthers pulled Lyon with 7:42 remaining, and Shesterkin had his chance to score the first goalie goal since Pekka Rinne did it with the Nashville Predators on Jan. 9, 2020. His uncontested shot from own crease went wide left.
"Before that shot I felt I would get a chance, but 'Lindy' [defenseman Ryan Lindgren] told me I cannot score before him," Shesterkin said, laughing. "Maybe next time a lucky bounce."
Chytil did score into the empty net at 16:49 for the 6-2 final.
NOTES: Bennett was injured crashing into the half-boards by the visitors bench in the final minute of the first period. He returned midway through the second but did not play in the third. Maurice said it's a lower-body injury and he doesn't expect Bennett to be available Tuesday. … Verhaeghe (four goals, two assists) and Lundell (three goals, three assists) extended their point streaks to five games. … The Panthers lead the NHL with 16 power-play goals since Jan. 1. … Zibanejad's first goal was his 250th in the NHL.