The Reds looked primed to leave New Zealand's capital with the three points after they played Wellington off the park and picked up two goals in a one-sided first half at windswept Sky Stadium.
But Airton Andrioli's side folded after the halftime break on Sunday as Wellington produced an impressive fightback to stay in touching distance of the finals places.
"It's very disappointing, especially the way it happened after playing such a controlled first half," Andrioli said.
"Everything that we put in place was working really well, and then coming back in the second half, showing that inconsistency was a little bit disappointing.
"I thought that we could have done better and shown a little bit more control.
"That's something that we need to have a look at, analyse and see why it happened."
Adelaide had eased into the lead with Ryan White converting a Panashe Madanha cross after 14 minutes.
And just before halftime Luke Duzel fired in from the edge of the box to put the Reds in command against a Wellington side who were cumbersome and lacking polish, failing to register a single shot on target through the first 45 minutes.
But Wellington began the second half like a team possessed and Giancarlo Italiano's men cancelled out the Reds' two-goal lead soon after the hour mark.
Centreback Isaac Hughes grabbed his first goal of the season when he nodded in a well-weighted Ramy Najjarine corner in the 57th minute.
Najjarine and Nigerian striker Ifeanyi Eze were asking plenty of questions of Adelaide from open play, and with the Reds on the ropes Wellington pulled level through Corban Piper.
Adelaide failed to deal with a marauding run down the right from 'Nix skipper Tim Payne, whose cross was turned into the back of the net by Piper for a 66th-minute equaliser.
"The intent in that first 10 to 15 minutes wasn't where it should be. We were pretty poor in the first half," said Italiano.
"In the second half, we changed a couple of things and executed really well."
Try as they might, Wellington were unable to find a winner, with midfielder Paolo Retre rattling a long-range shot off the crossbar.
Phoenix youngster Luke Brooke-Smith found the back of the net in added time, but his goal was immediately ruled out for offside.
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