Craig Anderson made 40 saves for Florida, which rebounded from a loss Tuesday to Boston. Thursday's game concluded the four-game season series, which the Panthers won, 3-0-1.
Markus Naslund had the lone goal for the Rangers, who remained winless under new head coach John Tortorella. He replaced Tom Renney on Monday, but New York lost in Tortorella's debut Wednesday, against Toronto.
The Rangers have lost four in a row overall and 12 of their last 14 games. Henrik Lundqvist gave up both goals on 22 shots in the loss.
Trailing in the third period, Florida got a pair of quick goals to overtake the lead.
The first came from Booth, who fired a backhander from the right circle between Lundqvist and the right post with 8:10 left in the game. One minute later, Horton one-timed a pass from behind the net past the Rangers goaltender.
The Rangers applied pressure in the final minute, but Anderson kept the puck out of the net. He stopped a turnaround chance with about 30 seconds left, then held onto the puck after a point blast with 15.3 ticks on the clock, and the Panthers held on for the win.
Naslund gave New York the lead 7:34 into the game, and that lasted to the end of the first and through the second period, during which Anderson made 13 saves and Lundqvist had eight.