The visitors broke the deadlock just after the half hour on Friday night and defended resolutely as the Reds threatened the Phoenix goal relentlessly.
Adelaide was eventually rewarded 10 minutes into the second half when, after a host of opportunities, Luka Jovanovic found the back of the net.
The Reds dominated the remainder of the second half but failed to find a winner.
The result extended Adelaide's unbeaten run to four while Wellington's winless run moves to seven, but they at least rose from the foot of the table.
Reds' coach Airton Andrioli said the draw was better than a loss.
"On a different day maybe we score and win the game," he said.
"I'm happy we've not lost the game, but wanted the three points tonight."
United almost opened the scoring within 15 seconds when Ethan Alagich released Jonny Yull with a long ball behind Wellington's defence and Luka Jovanovic headed just wide from Yull's whipped cross.
In the third minute, Adelaide almost gifted the Phoenix a goal when a mix-up between Pana Kikianis and Joshua Smits left the Reds' shot stopper scrambling to prevent a corner. In doing so, he kept the ball in for Ifeyani Eze, who squared for Ramy Najjarine, only for his shot to go just wide.
On the quarter hour, Juan Muniz bent a ball around the Phoenix defence for Yaya Dukuly who took it past Joshua Oluwayemi. With an open goal beckoning, Dukuly struck the upright from a tight angle.
For all United's dominance, it was Wellington who took the lead on 35 minutes.
Kazuki Nagasawa looped a cross into the Reds' penalty area from a free kick out wide on the right. Corban Piper rose highest and headed the ball into the net.
Smits managed to get a hand to the ball but couldn't keep it out. Replays showed Piper's header bounced off team-mate Manjrekar James on the way through.
United burst out of the sheds after the interval and turned up the pressure on the Phoenix. Jovanovic had a couple of half chances before eventually finding an equaliser on 55 minutes.
Yull drove towards the penalty area and threaded a pass through for Jovanovic, who sold a dummy to Oluwayemi and cut inside. The Reds' No.9 had to wriggle his way through the Wellington penalty area to find space, before blasting with the left foot into the roof of the net.
The home side continued to pepper Wellington's goal in search of a winner, testing Oluwayemi on multiple occasions.
Muniz picked out Ryan White inside the area with a delightful no-look pass, but the Phoenix shot stopper kept out White's toe-poke with his legs.
Wellington coach interim coach Chris Greenacre was pleased to have earned his first point since taking charge.
"The guys have given everything and with such a quick turnaround from our game against Sydney, I think if we would've been offered a point early in the week we would've taken it," he said.
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