Khabibulin didn't allow a goal after the Coyotes' Matthew Lombardi scored midway through the second period, which gave Phoenix a 2-0 lead at the time.
But Jordan Eberle halved the deficit with a tally later on in the middle frame and Teemu Hartikainen notched his first of the season for the equalizer 8 minutes into the third period.
After a scoreless overtime, the contest went to a shootout. Sam Gagner and Eberle beat Mike Smith, while Khabibulin stopped Mikkel Boedker and David Rundblad, securing his first win as a starter this season.
The victory snapped a two-game skid for the Oilers.
Kyle Chipchura scored and Smith ended with 41 stops for Phoenix.
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