Martin Straka and Michal Rozsival also scored for the Rangers (22-17-4), who lost the services of their leading scorer Jaromir Jagr in the second period due to a slight groin pull.
Mark Streit, Tomas Plekanec and Sheldon Souray scored for the Canadiens (23-13-5), who lost their second straight and their fourth in five games.
With the game tied 3-3, the Rangers got the winner at 13:55 of the third period after Brendan Shanahan beat Janne Niinimaa to a loose puck in the Montreal zone and fed the trailing Cullen, who beat David Aebischer with a wrist shot just under the crossbar for his ninth goal of the season.
Henrik Lundqvist made 21 saves to earn his 18th win of the season for the Rangers, while Aebischer made 30 stops for the Canadiens.
Souray's goal was his 11th this season on the power play, tying a Canadiens record for defencemen set by Guy Lapointe in 1974-75 and matched by Mathieu Schneider in 1993-94 and Patrice Brisebois in 2000-01.
The Canadiens, suffering from a flu bug, were without the services of captain Saku Koivu and rookie winger Guillaume Latendresse. They were replaced by Maxim Lapierre and Mikhail Grabovski, who were both called up from the Hamilton Bulldogs of the American Hockey League on Friday evening.
Grabovski, making his NHL debut, started the game centring a line with Alex Kovalev and Sergei Samsonov and had what looked to be his first career goal waved off by referee Marc Joannette at 8:50 of the second period.
Also scratched for the Canadiens were Garth Murray (hand) and Aaron Downey (healthy) as head coach Guy Carbonneau had defencemen Mark Streit and Mathieu Dandenault dressed as fourth-line wingers.
Centre Radek Bonk left the game early in the second period and did not return, further limiting Carbonneau's options up front.
Jagr, who entered the game fourth in NHL scoring, played sparingly in the second period and did not come out for the third.
With the game tied 2-2, the Rangers started the third on a power play and capitalized 38 seconds in when Rozsival's point shot beat a screened Aebischer for the defenceman's fifth of the season.
But Montreal tied it 3-3 on the power play at 8:40 of the third when Souray hammered a slap shot to the top corner for his 14th of the season.
The Rangers went ahead 1-0 at 10:20 of the first period when Cullen took a Fedor Tyutin feed at the Canadiens blue-line, cut between Craig Rivet and Francis Bouillon before picking the top corner on Aebischer.
New York took a 2-0 lead on Straka's 22nd of the season, but the Canadiens came back with two power-play goals to go into the third tied 2-2.
Streit scored his third on a one-timer from the point at 10:25 of the second while Plekanec scored his sixth of the season by converting his own rebound at 14:14 of the second.
Notes: Souray is seven short of the NHL record of 18 power-play goals by a defenceman set by Denis Potvin in 1974-75 and matched by Adrian Aucoin in. . . . The Canadiens play another afternoon game Sunday at 3 p.m. when they host the New Jersey Devils. . . . The afternoon game drew a sellout crowd of 21,273. . . . The Rangers were missing D Karel Rachunek (foot).