Ed Jovanovski scored in regulation for Phoenix, which won for the fourth time in five games. Tuomo Ruutu, Chicago's fifth shooter, was the first player to connect in the shootout. But Zigomanis scored in the bottom of the frame. After Chicago's Radim Vrbata was stopped by Phoenix goalie Michael Tellqvist, Doan beat Nikolai Khabibulin with a sweeping move.
Karl Stewart had a short-handed goal in regulation for Chicago. The Blackhawks are 3-0-2 under Denis Savard, who took over as head coach on Nov. 27.
Chicago's Khabibulin made 34 saves.
Tellqvist made his second start in goal for the Coyotes - and posted his second win - since being acquired in a trade from Toronto on Nov. 28. He stopped 30 shots.
The Blackhawks failed to score a power-play goal for the ninth straight game, going 0-for-6 to extend their slump with the advantage to 0-for-33. They wasted a one minute, five-second 5-on-3 chance early in the second period, and a 52-second 5-on-3 late in the third.
Stewart scored the lone goal of the first period on a short-handed breakaway with 6:51 left. Bryan Smolinski flipped a backhand pass up to Stewart, who skated in alone from center ice and beat Tellqvist high on the stick side.
Stewart's goal was his second in 18 NHL games, his first short-handed.
Jovanovski tied it 1-1 with 6:06 to go in the second from a faceoff in the Chicago zone. After Yanic Perreault won the draw, Jovanovski's rising shot from the left point slipped through traffic and beat Khabibulin high on the glove side.
Tellqvist made a point-blank save on Chicago's Mikael Holmqvist with 5:40 left in the third to maintain the tie. He also stopped Tony Salmelainen close-in with 2:43 left.
Chicago wasted a brief power-play chance in overtime.
Notes: Chicago F Martin Havlat may return from an ankle sprain in the Blackhawks' next game, Saturday at Minnesota. Havlat posted seven goals and six assists in the Blackhawks' first seven games this season, but hasn't played since suffering the injury Oct. 20 in Dallas. ... Chicago D and captain Adrian Aucoin missed his sixth game with a pulled groin and is day-to-day. ...Phoenix C Mike Ricci was a healthy scratch. Coyotes C Patrick Fischer returned Thursday after missing the team's 3-0 loss in Dallas on Wednesday with a groin injury.