The Rangers (4-3-3) led 2-0 in the first period but squandered it to the Sharks, who were trying to complete a sweep of a six-game road trip.
Artem Anisimov also scored in the second after Ryan McDonagh and Derek Stepan had goals in the first. Martin Biron made 24 saves to improve to 2-0 as Henrik Lundqvist's backup. Callahan added an assist, and Dan Girardi, Erik Christensen and Brandon Dubinsky all had two assists.
All that, and fan favourite Sean Avery could be back in the lineup Thursday if he clears waivers Tuesday.
Each team scored a goal in the game's final minutes of regulation.
Fleischmann's second goal came on a power play with 2:10 left and gave Florida a 3-2 lead. His shot from between the circles went over Pavelec's shoulder.
The Jets tied it 3-all at 19:10 on Kane's second goal on a shot from in front. The goal was awarded after a video review to determine if there was a high-stick.
Stalberg rushed down the right wing on a 3-on-2 break, but pulled up with a move that made Nashville defenceman Jack Hillen turn the wrong way. Stalberg then fired a shot past Pekke Rinne for his third goal in two games.
Chicago's Duncan Keith, the 2010 Norris Trophy winner as the NHL's top defenceman, left the game in the second period with an upper-body injury and didn't return.