The Coyotes got the game-winner when an Avalanche clearing pass around the boards struck an official's skate, causing it rebound directly to Phoenix's Kyle Chipchura inside the Avalanche zone.
The Coyotes center then whipped a pass to Doan, who was all alone in the slot to one-time a shot past a surprised Semyon Varlamov for the decisive tally.
Doan also scored in the first period and notched an assist while Oliver Ekman-Larsson added a score for Phoenix, which improved to 4-1-1 in February.
Mike Smith turned away 41-of-43 shots to earn the victory.
Mark Olver and Matt Duchene each netted a second-period goal and Varlamov finished with 27 saves for the Avs, who dropped a third straight.
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