Joffrey Lupul scored twice and Dan Sexton added a goal as the Ducks scored three times in a 9:51 span of the third period to force overtime.
Sbisa's goal came with Anaheim (2-1) on a two-man advantage after two Phoenix penalties, one in the final seconds of regulation.
Phoenix outshot the Ducks 46-21 overall and 38-6 in the first two periods.
Coyotes coach Wayne Gretzky, whose $8.5 million contract for 2009-10 is part of a bankruptcy dispute being adjudicated in a Phoenix court, missed his fourth straight game. Associate coach Ulf Samuelsson ran the team.
Mueller scored on a slap shot from just inside the blue line on a power play to give the Coyotes a 1-0 lead with 9:03 remaining in the first period.
Hoggan made it 2-0 53 seconds later, when he beat Timo Peilmeier to the far corner from the right circle.
Phoenix outshot Anaheim 30-2 in the first period, and the final Ducks' shot came from center ice as time expired.
Vandermeer scored from outside the right circle to give Phoenix a 3-0 lead with 18 seconds remaining in the second when new goalie Jean-Philippe Levasseur was screened by Brett MacLean.
Lupul scored Anaheim's first goal on a power play with 9:51 remaining during a 5-minute major and a game misconduct penalty on defenseman Francis Lessard.