The Coyotes never trailed in the contest, but saw a one-goal lead disappear three times before finally getting the game-winner from Whitney.
Phoenix will host the second game of the best-of-seven series Sunday.
Rostislav Klesla put Phoenix back in front early in the second, only to have Andrei Kostitsyn deadlock the game again midway through the period.
The Coyotes regained the advantage with 3:33 left in the second on a goal from Mikkel Boedker. Nashville was able to tie it with 4:42 left in regulation on a power-play goal from Martin Erat.
Mike Smith stopped 39 shots for the Coyotes, who are trying to reach the Western Conference finals for the first time in the history of the franchise. Rinne made 24 saves.