Steve Sullivan also scored in the shootout for Phoenix, which has won three of four.
The teams traded four goals over a six-minute stretch of the third period.
Perry had given the Ducks a 3-2 lead early in the period when he redirected Getzlaf's shot from the left circle past Smith. But Doan answered 11 seconds later, taking a pass from Kyle Chipchura and beating Jonas Hiller from just inside the right circle.
Again the Coyotes answered quickly, with Lombardi redirecting Matt Stone's shot between Hiller's pads 78 seconds later.
Chipchura had the best opportunity in overtime, skating behind the Ducks' defence to take a long outlet pass with just over a minute left. Hiller made the pad save.
Beleskey opened the scoring midway in the first period, slipping a shot between Smith's pads. Vermette tied the game 54 seconds into the second, putting home a long rebound off Hiller's left pad into the vacated left side of the net from the top of the crease.
Phoenix took its first lead on Yandle's blast from the top of the left circle. Getzlaf evened the score 2-2 with 7.8 seconds to go in the second period, taking a pass from Perry after the right wing stole the puck from Yandle in the neutral zone.
Hiller made 36 saves, while Smith finished with 27.
NOTES: Doan's 324th career goal moved him past Keith Tkachuk into second place on the franchise's career list, trailing only Dale Hawerchuk. Doan also became the third player in Winnipeg/Phoenix history to score 800 career points. . Doan has 30 goals and 66 points in 90 career games against the Ducks. . Perry has 36 points in 43 games and Getzlaf has 45 points in 44 games against Phoenix. . Lombardi has 100 career goals in 518 games. . Vermette's goal was his 150th in 644 career games. . The teams complete their three-game series Wednesday night in Anaheim. . Coyotes C Chris Brown made his NHL debut. . Phoenix C Martin Hanzal returned to the lineup for the first time since Feb. 18. Hanzal had an upper body injury.