The Rangers fell behind 3-2 on a goal by Calgary's Sean Monahan 4:44 into the final frame, but Chris Kreider notched the equalizer with 7:53 remaining in regulation.
After a scoreless overtime, the game moved to a shootout.
The Rangers took leads in the first, fourth and sixth rounds, but Calgary answered with scores each time -- including goals from Lee Stempniak and Paul Byron in the late going -- to keep the tiebreaker alive.
Pouliot, however, put New York ahead in the seventh on a dazzling play in which he moved all the way to the left but reached back to the right and slid the puck past Flames goalie Karri Ramo.
Henrik Lundqvist then stopped Mikael Backlund to secure the win for New York, which broke a five-game losing streak.
Derek Stepan finished with a goal and assist for New York.
Carl Hagelin also scored and Lundqvist made 18 saves.
Curtis Glencross supplied a goal and an assist for Calgary.
Backlund added a goal and Ramo finished with 29 saves.