The Rangers were trailing 2-1 when the Islanders were penalized for having too many men on the ice 4 1/2 minutes into the final period, and McDonagh responded with a power-play tally 30 seconds later.
Then with 6:14 remaining, the Islanders turned the puck over at center ice and the Rangers broke in on a 3-on-2 rush. Carl Hagelin fed a pass into the high slot and Pouliot sent a forehand shot past Evgeni Nabokov for the game-winner.
Chris Kreider also scored and Derek Stepan supplied two assists for the Rangers, who were coming off a 2-0 loss to Montreal in Monday's reopening of the renovated Madison Square Garden.
Cam Talbot stopped 22-of-24 shots.