After Johnson tied it with a power-play goal at 10:41 of the first, Palat scored a rebound goal from the left circle to make it 2-1 with 3:55 left in the period.
Nash snapped his seven-game goal drought to put the Rangers up 1-0 just 4:30 into the game. Derek Stepan set up the goal after stealing a pass by Lightning defenseman Radko Gudas.
Zuccarello got the Rangers even at 2 during a power play when his pass into the low slot deflected off Gudas into the net at 18:25. The center assisted on goals by Kreider and Pouliot.
"I don't think I've seen us mismanage the puck more," Tampa Bay coach Jon Cooper said. "That was painful to watch, to be honest."
NOTES: Lightning star Steven Stamkos, out since breaking his right leg on Nov. 11, did a series of agility drills during a 30 minute on-ice workout. The center, who hopes to return in February, resumed skating 2 1/2 weeks ago. ... Rangers RW Ryan Callahan (sprained knee) had a limited workout, wearing a no-contact jersey, during the team's morning skate. ... Tampa Bay D Eric Brewer (upper body), Sami Salo (upper body) and LW Ryan Malone (broken ankle) could return during a four-game trip that starts Wednesday at Vancouver. ... New York D Marc Staal (concussion) and D Anton Stralman (left hand and wrist) practiced. ... Tampa Bay D Dmitry Korobov assisted on Filppula's goal for his first NHL point. ... Rangers D Conor Allen played nearly 16 minutes in his NHL debut.