Montreal's Jaroslav Spacek tied the game 1-1 with about a minute left in the second period and Pouliot added the winner 6:03 into the third with a shot from low in the left circle that eluded Rangers goaltender Henrik Lundqvist.
"I should have had it," Lundqvist told reporters. "He took a real good shot, but it doesn't matter from that angle. I should keep my chest up a little more - it probably would have hit my shoulder."
It was the second straight come-from-behind triumph for the Canadiens, who got two goals in the final minutes of regulation before beating Boston 3-2 in overtime on Saturday.
With their third straight win, the Canadiens (24-16-3) remained two points behind the Bruins for first place in the Northeast Division.
Brandon Dubinsky had given New York (25-16-3) the lead when he opened the scoring about four minutes into the game, but the Canadiens ultimately got the best of Lundqvist, who finished with 36 saves.
Winger Wojtek Wolski, acquired from the Phoenix Coyotes on Monday for defenseman Michal Rozsival, nearly tied the game with under three minutes to play but his shot went off Montreal goalie Alex Auld's pads before sliding across the crease.
"Right then and there I kind of knew we were meant to win the game," said Auld, who finished with 25 saves. "The bounces seemed to be going our way."
(Editing by Frank Pingue)