Tom Doyle headed Alex Rufer's corner back across goal and Adelaide keeper Paul Izzo punched the ball out to the edge of the penalty area, where Mandi bounced it back over him and into the net.
During a first half of few chances, the next best two came from free kicks on the edge of each penalty area: Phoenix keeper Filip Kurto did well to save from Isaias, and Izzo blocked a deflected effort from Mandi.
As the first half meandered to a close, the game was going according to plan for the Phoenix — right up until Adelaide equalised.
A Ben Halloran cross cleared three jumping players and the ball struck a grounded Doyle in the head and bounced into goal amid an eerie silence at Westpac Stadium. The Phoenix have now gone 17 home games with keeping a clean sheet.
Just before or after halftime is always a psychologically great time to get a goal and Adelaide achieved both, taking the lead two minutes after the restart.
Ken Ilso, who admitted after the game he'd headed only a couple of goals his entire career, got between Michal Kopczynski and Stephen Taylor to plant a header into the bottom corner off a superb Michael Marrone cross.
The game had turned in a matter of minutes and when Adelaide netted a third, Phoenix hopes were effectively dashed, even with 25 minutes still to play.
Wellington were the team supposed to threaten on the break but instead Adelaide were more dangerous on the counter-attack.
That's how they scored their decisive third goal, when Ryan Kitto curled in a tantalising cross and the ball ended up nestling in the back of the net.
Doyle looked to have got a touch on it but was spared the ignominy of a second own goal when officials credited the effort to Ilso.
A frustrated Rudan shattered the side of his dugout in what proved the most effective blow landed by anyone in Phoenix colours in the second half.
Wellington never looked like getting back in the match and Adelaide squandered a couple of good chances to add a fourth.
The win extended Adelaide's unbeaten run against the Phoenix to nine games and will only deepen the worry lines for Wellington fans.
"We're working hard but results just aren't going our way," said Phoenix attacker David Williams. "It's disappointing; a few key instances let us down but that's football and it can be a long season if we don't address that. We've got to be better in certain situations."
Adelaide United 3 (T. Doyle own goal 45+1, Ken Ilso 47, 64).
Wellington Phoenix 1 (Mandi 14)
Halftime: 1-1.
