In an uninspiring contest the visitors were made to work hard for the three points with Marcelo Carrusca's penalty enough to get them over the line.
Marcelo Carrusca's 58th-minute penalty lifted Josep Gombau's men into fourth on the A-League table, just two points behind the second-placed Central Coast with two rounds remaining.
The loss all but ended the Phoenix's finals chances and they will officially be out of the running if Newcastle beat Perth later on Sunday.
Carrusca's spot-kick ensured Adelaide stayed unbeaten at Westpac Stadium, improving their record to two wins and four draws.
But the penalty – won by Tarek Elrich – was one of few decent chances in the New Zealand capital.
Kenny Cunningham had no hesitation in testing Eugene Galekovic early, but the Costa Rican's 25-yard strike was straight at the goalkeeper.
It was a largely frantic start with neither team settling, while Gombau's men were unable to keep possession for long periods.
With Shaun Timmins yellow-carded for a rash challenge on 12 minutes, Adelaide targeted the left-back by attacking through Fabio Ferreira during the first half.
As the visitors struggled to get going, Cunningham tried from range again on 19 minutes, but his cheekily attempted chip floated over.
Ferreira got behind Timmins on a couple of occasions and only a final pass was lacking, likewise for Carrusca when he was released down the right.
Stein Huysegems had a chance to finish a rather forgettable first half with the game's opening goal just before the break.
The Belgian was played through, but his poke was kept out by Galekovic's strong right hand in a one-on-one.
Largely unseen throughout the first half, Bruce Djite and Michael Zullo were more involved early in the second.
The opener came after a move down Zullo's left flank as Elrich and Carrusca combined, with the latter producing a delightful flick in their one-two.
Elrich took a touch too many but drew a sloppy sliding challenge from Andrew Durante, who gave away the spot-kick.
Carrusca stepped up and produced a near-perfect 'Panenka' that went in just under the crossbar as Glen Moss dived to his right.
Jeremy Brockie almost responded immediately as his powerful shot was well saved by Galekovic, but Adelaide should have sealed the points with a quarter of an hour remaining.
Zullo was released down the left and the on-loan Utrecht attacker failed to pick out Carrusca when shooting from an angle was perhaps the better option.
Michael Boxall volleyed a good chance well over soon after and Adelaide held on despite a couple of nervous moments late in the encounter.
Wellington Phoenix 0
Adelaide United 1 (Carrusca 58p)
Crowd: TBA @ Westpac Stadium