Chris Summers got his first NHL goal and Rob Klinkhammer also scored to put Phoenix up 2-0 after one period. Calgary got power-play goals from Curtis Glencross and Mikael Backlund to tie it at 2 after two.
After a frustrating series of power plays, Rahim Vrbata rocketed a shot toward the net and Doan deflected it past Joni Ortio for the deciding goal with 11:11 to play.
The Coyotes pulled into a tie with the Western Conference's No. 8 team Dallas with 75 points, but the Stars have played two fewer games and hold the tiebreaker.
The Coyotes' Mike Smith, strong all night, had 30 saves, while Ortio stopped 22 shots.
Phoenix had five power plays through 10 minutes of the first 13:13 of the third period.
Sutton scored on a slick deflection of Lauri Korpikoski's pass just 4:35 into the game.
Phoenix made it 2-0 with 9:43 left in the opening period thanks to a miscue by Ortio.
Ortio, recalled from Abbotsford of the American Hockey League on Feb. 3, and playing in his seventh NHL game, ventured behind the net to try to clear the puck but lost control of it. It went right to Klinkhammer, who tapped it in to the empty net for his 10th goal.
Calgary cut the lead to 2-1 on a power-play goal by Glencross 7:46 into in the second period. With Derek Morris in the penalty box for holding, Glencross positioned himself directly in front of the net, and glanced Mike Cammalleri's sizzling shot past Smith.
Phoenix lost an opportunity for a full power play when the Coyotes' Antoine Vermette was called for holding just 44 seconds into Cammalleri's penalty for interference.
Another power-play goal tied it for Calgary moments later. Blacklund scored from a crowd in front of the net and it was 2-2 with 4:23 left in the second period. The goal came with Phoenix's Jeff Halpern serving a penalty for tripping.
The Flames nearly got another goal in the final seconds of the second when Cammalleri got the puck point-blank in front of the net, but Smith was able to knock the shot away with his right pad.
Phoenix's sixth and final power play of the third period ended with 12 seconds to play.
On Friday night, Calgary trailed at Dallas 3-1 with eight minutes to play in regulation before rallying to force overtime, then win in the shootout 4-3.