Vermette found a loose puck in front of Senators goalie Craig Anderson and pushed it through to the back of the net at 2:23. He now had three goals against his former team in five games.
Radim Vrbata also scored for the Coyotes (19-10-6), who halted a three-game losing streak.
Both goaltenders, Anderson for the Senators and Mike Smith for the Coyotes, saw plenty of action as Ottawa outshot Phoenix 40-38.
Smith made solid back-to-back saves pad saves on Kyle Turris and Clarke MacArthur two minutes in overtime to keep the game going. The second, on MacArthur, Smith was on his stomach and just stretched his right leg out enough to get his toe on the puck.
Seconds later Vermette scored the winner
Vermette's second goal tied the game with just 2:13 to play in the third period and forced the overtime session.
While on the power play the Coyotes were aided with two horrible clearing attempts by the Senators and eventually Vermette took a pass from David Moss and deposited the puck into an empty net behind a fallen Anderson.
Ottawa was on the positive side of a 2-0 score early in the second period.
The lead was short lived though as the Coyotes scored twice in a little more than three minutes, including once on the power play, to tie the game.
Vermette scored his first goal when he beat Anderson from the slot at 5:28. Vrbata then scored on the power play at 8:47 when he moved in from the point and beat Anderson short side with a shot that barely made it past teammate Martin Hanzal.
Methot then gave the Senators the lead back with his goal at 14:51.
Notes: Patrick Wiercioch, Joe Corvo and Mika Zibanejad were scratches for the Senators Saturday while the Coyotes elected to sit David Rundblad, David Schlemko, Paul Bissonnette and Jeff Halpern…Saturday's game was the first time Phoenix played in Ottawa since Oct. 26, 2010 when they lost 5-2…Senators defenceman Cody Ceci turned 20 on Saturday…Erik Karlsson of the Senators, lead NHL defenceman in scoring with 34 points…The Phoenix Coyotes are the only NHL club this season the hasn't scored a short-handed goal.