Pouliot, who also registered an assist on a second period goal by Carl Hagelin, had a nifty backhand move in the shootout, and scored past Flames goaltender Karri Ramo.
Rangers center Derek Stepan led New York in scoring with a goal and an assist. With the win, the Rangers improve to 16-17-1, and are four points back of the Toronto Maple Leafs for the eighth and final playoff spot in the Eastern Conference.
Despite losing to the Rangers, the Flames did pick up a very valuable point in the Western Conference. However, they are still 12 points back of the Minnesota Wild for a playoff spot.
The Flames should be pleased with the fact that their offense was rolling. They did score three times each in regulation and the shootout on 2012 Vezina Trophy winner Henrik Lundqvist.
The Flames next play Tuesday in Boston.