Perth are three points outside the top six in eighth-place, with two matches left to play.
Playing at Eden Park in front of a bumper crowd of 22,223 fans, the Phoenix thought they had secured an early penalty when Tim Payne went down inside the box under pressure from Kosuke Ota in just the second minute.
Referee Alex King awarded the spot kick but after a lengthy VAR review reversed his decision.
The home team did hit the front midway through the opening half however when Israeli striker Tomer Hemed found space inside the box and slammed home Louis Fenton's cutback for his 10th goal of the season.
The Glory's comeback was then complete 14 minutes later when youngster Callum Timmins curled home a sensational 25-metre effort for his first A-League goal.
The Phoenix weren't done however with Davila coming off the bench to mark his return from a hamstring injury and having an instant impact when he found Jaushua Sotirio and the Wellington winger stroked the ball past Reddy from the edge of the area to equalise.
Wellington then surged home for a winner in the dying minutes and thought they'd done enough when Devlin won his penalty only for Reddy's heroics to deny the home team.
In a furious closing few minutes, the Phoenix poured forward and had multiple appeals for another spot kick but King waved them all away before ending the contest.
Perth now return to Australia to face Brisbane Roar on Wednesday in a must-win clash for their finals aspirations while the Phoenix simply have to beat Macarthur in their last regular season fixture next Friday and hope other results go their way to make the top six.