Atlanta had won just once in its eight previous games, but the two points they picked up against the Rangers moved the Thrashers in front of idle Carolina for the eighth and final playoff position in the Eastern Conference. New York has lost six straight.
It was a 1-1 game heading into the third, but New York moved in front 71 seconds into the period on a goal from Brandon Dubinsky.
Kane tied it with 5:51 to play and then scored a power-play goal against Henrik Lundqvist.
Ondrej Pavelec got the win by making 27 saves. Lundqvist stopped 32-of-35 shots. Atlanta has defeated the Rangers in seven of their last nine meetings.