The teams put on an entertaining overtime period in which they traded failed scoring chances. Toronto's Tyler Bozak missed on a breakaway, and Coyotes veteran Ray Whitney looked skyward after Ben Scrivens robbed him with a glove save.
Mike Komisarek and Kessel had goals in the third period for the Maple Leafs (10-6-2), who have dropped four straight home games.
The Coyotes improved to 7-1-2 in their last 10 games, using a defensive-minded game plan that they strayed from in the third period and overtime.
"We try to take away the Grade-A scoring opportunities," Doan said. "Mind you, when they've got Phil Kessel, he's going to get a few chances going down the wing."
Kessel had one of his best games of the season, logging a game-high 26 minutes, 32 seconds of ice time and looking dangerous virtually every shift he took. The only thing he failed to do was beat Mike Smith in the shootout.
"We battled," Kessel said. "We really tried to win this game and we battled back. Hopefully we can build on it."
Scrivens had a tough time in the opening minutes. Doan gave Phoenix a 1-0 lead at 1:54 after linemate Mikkel Boedker stole the puck from Maple Leafs captain Dion Phaneuf as he fell to the ice. He got the puck to Doan, who scored with a wrist shot that went in off Scrivens.
Ekman-Larsson doubled the lead at 5:18 with a long wrist shot from the point that found its way through traffic.
The Maple Leafs sprung to life when Komisarek ended Smith's shutout bid with his first goal since October 2010. The defenseman jumped in off the point and converted a pass from Matthew Lombardi at 2:08 of the third period.
That was soon followed by Kessel's NHL-leading 13th goal on a power play. Kessel sneaked in off the boards and slammed home a rebound to make it 2-2.
Scrivens ensured that was enough to get the game to overtime by coming up big on a penalty kill in the final five minutes. He then got his glove on Martin Hanzal's shot.
"We've got to bring that same resilience to start the game," Komisarek said. "Not sort of feeling the other team out and seeing what kind of game it's going to be. We need to dictate the play and assert ourselves."
Toronto finished the game without two-thirds of its second line. MacArthur didn't return for the third period because of an undisclosed lower body injury. Grabovski hobbled off with a few minutes to play following a collision at center ice.
Their status for Thursday at Nashville is unknown, but the Maple Leafs expect to have center Tim Connolly back in the lineup.
Phoenix is 6-0 against the Leafs since the lockout, outscoring them 29-13. ... Canadian tennis player Milos Raonic took in the game. He will face Pete Sampras in an exhibition match at Air Canada Centre on Thursday night. ... Announced attendance was 19,522.