With the sides struggling to break the deadlock deep into their Saturday clash, Old settled the tie with a sweeping right-foot strike that flew past goalkeeper Thomas Heward-Belle in the 79th minute.
Celebrations were briefly cut short when the offside flag was raised, but concerns were quickly allayed as a VAR check confirmed the goal.
"It was a bit of a sucker-punch that the goal got allowed because it looked offside," United coach John Aloisi said after the match.
"The camera angles aren't that good here because of the way the stadium is and there's not as many cameras as normal.
"You can't change the decision, so there's no use me getting angry.
"You just have to get on with it and make sure that the players aren't getting too down.
"They couldn't believe it but we have to move on. We still had chances after that to go and equalise."
Phoenix manager Giancarlo Italiano said he also initially believed Old was offside, but welcomed the three points.
"I didn't envisage that we'd be top of the table after six rounds," Italiano said.
"But I don't think we were in complete control today, even though the stats may have suggested we probably had the better chances.
"The fact we were able to grind and get that result, it's promising.
"I'm just happy we came away with three points."
After three straight home defeats to start the season, United were keen to bury their Mars Stadium hoodoo, pressing high and attacking aggressively from the opening whistle.
Angus Thurgate won a free-kick eight minutes in after a foul by Wellington's Kosta Barbarouses but the opportunity was squandered when Ben Garuccio delivered the ball straight into the hands of Phoenix goalkeeper Alex Paulsen.
United forward Lachie Wales, in his 150th ALM game, went close to breaking the deadlock but Paulsen was again a brick wall in front of the net.
Wellington then brought the heat, with Bulgarian midfielder Bozhidar Kraev and Barbarouses taking on the United defence in a bid to get on the scoreboard before the main break.
The visitors began the second-half in much the same fashion, with Tim Payne and David Ball bursting towards goal shortly after the restart.
Poor execution cost them, though, with Ball missing from point-blank range.
United's Noah Botic, Nikita Rukavytsya and Riku Danzaki hit back as the home side pressed, but Paulsen remained unconquerable.
With the game looking destined for a 0-0 draw, Old struck with 11 minutes remaining to condemn cellar-dwellers United to a fifth-consecutive defeat.
Italiano has now gone undefeated in his first six games at the helm.
12°C