Melbourne played more than 80 minutes a man down after Rhys Williams was sent off in the eighth minute, but Wellington were unconvincing despite their numerical advantage.
Trailing 1-0 at halftime, the Nix needed two second-half goals to sneak home for just their second win of the season.
Williams' over-enthusiastic tackle on Andrija Kaludjerovic earned only yellow initially, but was upgraded to red after the VAR ruled the central defender's foul had stopped a clear goal-scoring opportunity.
Coach Darije Kalezic made his intentions clear as the 20-minute mark ticked over, benching centre back Marco Rossi and injecting new signing Nathan Burns.
But Wellington couldn't capitalise on their numerical advantage, and paid the price in the 40th minute when Melbourne pounced on a combination of disorganised defending and poor keeping.
Former Phoenix favourite and All Whites striker Kosta Barbarouses found space down the right, running onto Jason Geria's through ball, bamboozling Tom Doyle then slotting home past Lewis Italiano's near post.
Playing a lengthy eight minutes of extra time, Wellington continued to attack and looked to have levelled through Kaludjerovic, but Roy Krishna was harshly ruled offside.
Trailing at the break despite having had 11 shots on goal to Victory's two, Wellington's desperation only increased in the second spell.
Kalezic boosted his midfield further in bringing on his second new signing, Serbian midfielder Matija Ljujic, after the break but the new combination again took time to settle.
The equaliser finally came in the 64th minute, after Thomas fumbled a stinging strike from Burns and Krishna pounced to tap home.
Burns also featured in Wellington's 83rd-minute winner, drilling a low cross into the box which took a touch from Melbourne defender Mark Milligan before dribbling across the line.
Wellington still remain mired at the bottom of the ladder, but will go into Saturday's match against Western Sydney Wanderers at Westpac Stadium a little more heartened.
