Whitney also started the rally when he scored an unassisted goal 8 minutes into the final period, cutting Phoenix's deficit to 2-1.
Raffi Torres then scored the equalizer with 2:23 left, forcing a scoreless overtime and then the shootout.
Whitney's backhander past Devan Dubnyk in the third round of the tiebreaker gave the advantage to the Coyotes, and Mike Smith made it hold up with a final save on Ryan Nugent-Hopkins.
Smith finished with 36 saves for Phoenix, which vaulted into seventh place in the tightly packed Western Conference standings with 83 points.
Swim Week Miami Beach 2014 [PHOTOS]