Phoenix struck first just 1:41 into the opening period when Doan grabbed a loose puck to the left of the net after a flurry of shots were blocked in front and sent a wrister into the twine.
New York, though, was issued a power play near the midpoint of the first and converted on the man advantage when Nielsen cleaned up a rebound to send the game into the first intermission tied at one.
But the second period belonged to Phoenix as the hosts tallied three goals in the frame.
The first came off the stick of Doan at the 1:39 mark on a breakaway and Pyatt gave the Coyotes a two-goal advantage two minutes later with a wrister from the slot that went past the stick side of Nabokov.
Ekman-Larsson lifted a shot into the top-right corner of the net from the left circle with 5:21 left in the second to increase the Coyotes' cushion.
Doan completed the scoring and his first career hat trick on the 308th goal of his 16-year career with one second to go in the third period on a slap shot from the top of the left circle that screamed past the goaltender just before time expired.