The Coyotes trailed 3-2 late in regulation when Ottawa's Erik Condra was called for a high-sticking penalty with 3:25 remaining and Vermette converted a pass a from David Moss for a power-play equalizer.
In overtime, Vermette won a face-off in the Senators zone, moved to the front of the net and batted in the rebound of a Derek Morris shot for the game-winner and his second career hat trick.
The former Senators center also scored in the second period as Phoenix logged its first victory in Ottawa since 2010.
Marc Methot netted two goals and Zack Smith also scored for the Sens, who dropped a third straight.
Craig Anderson made 34 saves.