Gabriel Landeskog netted the eventual game-winner and Matt Hunwick added a goal for the Avalanche, who entered the game clinging to the eighth and final playoff spot in the West.
The victory moved them into the seventh position, two points better than idle Phoenix with eight games to play.
The Rangers, meanwhile, saw their Eastern Conference and Atlantic Division lead over Pittsburgh dwindle to two points.
Henrik Lundqvist made 17 saves and Mats Zuccarello netted the only goal for New York.
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