The Reds scored twice in quick succession early in the first half and looked to be cruising before Dylan Pierias was given his marching orders on 36 minutes.
City pulled a goal back from the resultant free kick on Friday night and the hosts' lead looked shaky. But, despite playing almost an hour with a man down, United scored twice late in the second half to record a memorable victory.
Adelaide broke the deadlock on 12 minutes, capitalising on an error from the visitors.
Zane Schreiber's pass to goalkeeper Patrick Beach was short and Luka Jovanovic intercepted. He teed up Ethan Alagich whose first time effort from outside the area lofted over Beach and Liam Bonetig.
Three minutes later and United doubled their lead when Yaya Dukuly drove towards the byline and squared for Jovanovic, who controlled and shot with his second touch into the bottom corner.
The Reds were in the driver's seat and in search of a third before Pierias was shown a straight red card after a tangle with Andreas Kuen.
From the resulting free kick, City's number 10, Takeshi Kanamori, pulled a goal back with a contender for goal of the season.
Aziz Behich's cross struck the wall and the rebound fell to Kanamori who unleashed a first time left-footed volley from 25 metres which flew into the top corner leaving Joshua Smits no chance.
The home side thought they snatched a third on 68 minutes when Smits launched a long ball from the penalty area over City's defence for substitute Panache Madanha to chase.
The Adelaide speedster burst past City substitute Peter Antoniou who tripped in the tussle, allowing Madanha to charge towards goal and slot past Beach.
Referee Shaun Evans called the ball back for a foul on Antoniou, incensing almost everyone inside Coopers Stadium.
Adelaide would have the final say however, scoring twice more to kill off the visitors.
On 79 minutes Ryan White regained possession for United outside the area before driving towards goal. The substitute danced past two City defenders, cut inside and coolly dispatching past Beach at the near post.
Five minutes later and fellow sub Jay Barnett iced the stunning performance for United.
With City committed forward, the Reds broke free and it was Barnett who took possession just inside his own half before releasing White tearing through the middle.
White sprinted towards goal and with just Beach to beat struck the upright with his effort, but Barnett continued his run in support and knocked in the rebound.
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