Archie Goodwin nailed the winner in the 89th minute on Friday night after the Reds had raced to an early 2-0 lead before coughing up two goals in four minutes just after the half hour.
The victory snapped a run of eight games without a win for Adelaide and temporarily lifted them into sixth place, two points above Sydney FC, who meet Newcastle Jets on Sunday.
Reds coach Carl Veart said his overriding emotion was one of relief.
"At the moment I'm sitting here relieved that we won the game tonight because we made it difficult for ourselves," he said.
"We started so well and scored two good goals and could've easily had a couple more, and then to concede some fairly soft goals ..."
Adelaide made a dominant start and broke the deadlock in the eighth minute. Zach Clough looped in a corner deep into the penalty area where Panagiotis Kikianis speared a header between Wellington keeper Joshua Oluwayemi and the post.
The Reds doubled their lead just four minutes later thanks to Stefan Mauk.
Ethan Alagich exchanged a one-two inside the area with Clough, before lifting a delicate chip towards the penalty spot where Mauk was charging through to head into the net.
United continued their dominance and had two chances to make it 3-0. On 16 minutes, Dylan Pierias set off on a dazzling run, beating three defenders before seeing his curling effort saved by Oluwayemi.
On 20 minutes, Mauk struck the upright from just inside the area after he was teed up by Alagich.
The home side were made to rue their missed opportunities when the Phoenix pulled a goal back on 31 minutes - completely against the run of play, with their first touch inside Adelaide's penalty area, and first shot on goal.
Hideki Ishige outmuscled Ryan Kitto and held off the defender as he drove forward, slipping through Kosta Barbarouses. The Wellington veteran darted into the penalty area and poked the ball past Reds' keeper Max Vartuli.
Four minutes later the visitors drew level. Sam Sutton recycled an attack and launched a long ball towards the back post.
Scott Wootton outjumped Kitto and nodded it back into the six-yard area where Isaac Hughes burst to glance a header over Vartuli.
Wellington looked the better side in the second half and Barbarouses went close on 53 minutes, getting in behind the Reds defence and almost scoring past the on-rushing Vartuli, with the ball cleared off the line by Kikianis.
On 77 minutes, Adelaide substitute Jonny Yull almost scored from a corner, whipping in a deadly ball which was cleared off the line at the back post by Hughes.
Soon after, Yull forced Oluwayemi into a stunning save when he almost picked out the top corner from a 25 metre free kick. The Phoenix keeper was at full stretch and tipped the effort around the post.
The Reds eventually found a winner on 89 minutes through young striker Archie Goodwin.
Mauk clipped a ball into the area which was cleared by Wellington, but only as far as Pierias. The United full back controlled, and drilled low and hard into the six yard area where Goodwin was left unchecked to tap into the net.
Phoenix coach Giancarlo Italiano lamented giving the Reds a two-goal head start.
"The first 12 minutes, I'm just disappointed we gifted two early goals," he said.
"From a physical point, I thought we were slow to start and didn't show any energy at the beginning.
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