Wellington left New Zealand during the week for the game on a high following their 5-2 hammering of the Newcastle Jets a week ago and they had won three of their past four outings.
The home side stunned the Phoenix after 17 minutes when nippy striker Bruce Djite turned in a sharp shot to beat Wellington goalkeeper Tony Warner on his near post.
There was some crafty interplay in the lead up to Djite's goal and although Warner was beaten, he was hardly to blame as the shot came off the striker's boot like a bullet.
Wellington had their chances in the opening spell and Tim Brown was gifted the opportunity to equal the scores after 40 minutes when he was left unmarked from point-blank range but he failed to guide a well-placed free kick in to the back of the net and could only find the post.
After Manny Muscat committed a foul and was lucky not to receive his second yellow card in the 62nd minute, Dario Vidosic blasted a powerful free kick from the edge of the box; it was saved by Warner but youngster Antony Golec was first to pick up the scraps and hammered the ball home to double Adelaide's advantage.
Phoenix coach Ricki Herbert tinkered with his winning line-up from the victory over Newcastle and restored Tony Lochhead at left back after he recovered from a groin injury.
Lochhead's return meant Vince Lia left the back line and pushed in to the midfield and forced the versatile Alex Smith out of the squad.
Herbert may have regretted that decision when the final whistle blew as the creative Smith, who had impressed during his two recent starts, offers something different in the middle of the park to Lia and Brown.
Referee Chris Beath was kept busy in the second spell as the yellow cards were dished out with regularity as the benches were cleared during a game that was played in searing temperatures of up to 33 degrees.
The result lifted Adelaide from the foot of the ladder to seventh, while Wellington remain in fifth.
The Phoenix return to the comforts of home next week when they host Sydney FC on Wednesday and Perth Glory on Sunday at Westpac Stadium.