The Coyotes got the equalizer with 7 1-2 minutes left in the third period when Chad Kolarik took the puck away from defenseman Sheldon Brookbank right in front of the crease on the forecheck and quickly tucked it between Pielmeier's stick and the left post.
The Coyotes nearly won it in the first minute of overtime during a power play, but Peter Mueller's 45-foot wrist shot from the slot hit the right post after beating Pielmeier to the glove side.
Ryan tied it 1-all at 3:09 of the second period, converting a rebound of Dan Sexton's wrist shot. Ryan's second goal deflected into the net off the blade of Phoenix defenseman David Schlemko's stick just 68 seconds into the third period after he carried the puck through the left circle and veered toward the crease.
Lauri Korpikoski scored Phoenix's first goal on a deflection of Keith Yandle's long slapshot during a power play with 21 seconds left in the first period, after Steve Eminger was sent off for interference.
LaBarbera, who signed a two-year free agent contract with Phoenix in July, made 28 saves in his Coyotes debut.
Coyotes associate coach Ulf Samuelsson again called the shots from behind the bench while Wayne Gretzky's status as head coach remained in limbo. Gretzky -- also a minority owner of the team -- is awaiting a ruling by Judge Redfield T. Baum in the bankruptcy case of Coyotes owner Jerry Moyes, which is expected to be settled within 10 days.