Dallas' Jamie Langenbrunner was penalized for tripping with 19 seconds left in regulation, and Vrbata capitalized on the 4-on-3 advantage in the extra period with a wrist shot from low in the right circle.
Michal Rozsival had a power-play goal, Mikkel Boedker scored the tying goal in the third period, and Ilya Bryzgalov made 16 saves as the Coyotes won their third in a row.
Brad Richards had to wait a few anxious minutes before his 22nd goal of the season was confirmed on video review, and Kari Lehtonen stopped 32 shots for Dallas.
Steve Ott also scored for the Stars, 1-5-1 in their past seven.
Bryzgalov was coming off two consecutive shutouts for the first time in his career, but the Stars broke that streak when Ott fired into the upper right corner of the net from low in the left circle at 10:48 of the first period.
Phoenix tied it at 1 when Rozsival tapped in a rebound at 18:22 of the opening period.
Richards' rebound gave Dallas a 2-1 edge at 5:56 of the second period.
Teammate Nicklas Grossman took the original shot seconds after exiting the penalty box, and Grossman was pushed into Bryzgalov by a Phoenix defenseman as Richards took his shot.
The review was necessary to determine whether the puck crossed the goal line, and an inconclusive replay meant that the goal stood.
Richards also assisted on Ott's goal, giving him 700 career points.
Boedker tied it at 2 at 5:37 of the third period with his first goal of the season when he converted Ray Whitney's cross-ice pass.
Notes: Attendance was announced at 11,488, but half that number were in the seats due to icy roads from the area's second major storm in the past nine days. ... Dallas activated C Tom Wandell off injured reserve after missing four games due to a shoulder injury sustained on Jan. 24 at Vancouver. ... The Stars are still without C Adam Burish (broken orbital bone), RW Krys Barch (scratched cornea and fractured orbital bone), RW Ray Sawada (separated shoulder and broken nose), C Jamie Benn (shoulder) and C Toby Petersen (leg). ... Stars forward Tomas Vincour made his NHL debut. ... The Coyotes won their 500th game since moving to Phoenix in 1996. ... D Ed Jovanovski (lower body) missed his fourth straight game for the Coyotes, but C Vernon Fiddler (upper body) was back in the lineup after missing 10 games.