Kostitsyn was at the right side of the net when he collected a rebound off a shot from the top of the slot and deftly put it past Ilya Bryzgalov for the game-winner.
The Canadiens won for the fourth time in five games and collected their 12th straight victory over Phoenix.
Derek Morris netted an unassisted goal on a power play early in the third period to tie the game 2-2 and Kyle Turris also scored in Phoenix's latest loss to Montreal.
Bryzgalov turned aside 26-of-29 shots for the Coyotes.
Notable deaths of 2014 [PHOTOS]
Julius Randle breaks leg during NBA debut