Glory took the lead in the 16th minute through a stunning opener from Brazilian winger Sidnei before Stein Huysegems and Paul Ifill scored either side of half-time to give Wellington the lead.
The Phoenix looked set to record their first win over the season until Michael Thwaite levelled the scores from a Sidnei cross in the 60th minute and Adrian Zahra capitalised on a great run from veteran Steven McGarry in the final 10 minutes to secure three points for the home side.
Sidnei, who only minutes earlier appeared to injured his left hamstring, rounded off the win in injury time with his second of the evening when Glenn Moss fumbled what should have been a regulation save.
It was Glory's first win in five games and puts them back in the A-League's top six ahead of three more games this weekend.
Elsewhere, a double from Besart Berisha helped Brisbane Roar come from behind to record a 2-1 win at Adelaide United.
Adelaide grabbed an early lead through Osama Malik, who retained his place in the heart of defence with Nigel Boogaard left out of the side completely, before Berisha found the net in each half to ensure Brisbane claimed maximum points.
The home side went into the game searching for their first victory since the opening day of the season, but it was the Roar who secured a third successive victory that keeps them top of the standings.
