Phoenix snapped a three-game losing streak, after twice taking the lead, only to see Taylor Hall tie it both times for the Oilers.
Doan scored his eighth goal of the season to put the Coyotes in front for good and Oliver Ekman-Larsson, who assisted on Doan's goal, gave Phoenix a two-goal lead with 5:23 to play.
Both of Hall's goals came on the power play, the second one creating a 2-2 tie 6:11 into the third period.
That was the last goal surrendered by Mike Smith, who made 18 saves to snap a personal three-game slide. Smith has just two victories in seven December appearances.
Devan Dubnyk stopped 38 shots for Edmonton, which has dropped back-to-back games.