Vermette opened the scoring with a power-play goal in the second period and then notched the final goal in the third period for the Coyotes, who completed a 2-1-1 road trip.
Jordan Szwarz added the second goal for the Coyotes, who outshot Colorado in the second period 11-7.
Patrick Bordeleau scored the only goal for the Avalanche at 14:28 of the second period to cut the Phoenix lead to 2-1. Semyon Varlamov tallied 22 saves in the loss.
Colorado outshot the visitors 12-5 in the third period but could not get past Griess.