Phoenix opened the scoring in Monday's (AEST) match through Tomer Hemed's fifth goal in four games.
That well-taken strike was cancelled out by substitute Bruno Fornaroli's composed finish past Oliver Sail, before Callum Timmins completed the turnaround with a sensational long-ranger.
Perth was unable to hold on for a first away win in seven attempts, though, as another Wellington substitute in Jaushua Sotirio curled in an equaliser six minutes later.
There was still time for more late drama as Cameron Devlin went down under a challenge from Nick D'Agostino, only for Ulises Davila's resulting penalty to be kept out by Reddy.
Wellington is now unbeaten in a club-record 10 A-League games, but they are one point outside the top six having played a game more than Macarthur, while Perth are two points further back.
Brisbane Roar still has work to do if it is to seal a Finals berth, meanwhile, after losing 2-0 at Western Sydney Wanderers in the second of Sunday's matches.
Wanderers had lost three games in a row heading into the contest at Bankwest Stadium, but goals from Bernie Ibini and Bruce Kamau kept their slim top-six hopes just about alive.
Daniel Margush starred for the home side with six saves, including a couple of fine stops to keep out Riku Danzaki either side of Ibidi's 20-yard shot into the bottom-left corner.
The visitor pushed hard for an equaliser and had a penalty shout rejected when Jesse Daley went down inside the box under contact from Ziggy Gordon.
With players committed forward, Roar had a chance to pick off their opponents late on and Kamau made the most of his opportunity by chipping the ball over Jamie Young at his near post to seal the win.