The Roar looked destined to win after Massimo Maccarone opened his account just eight minutes into his home debut, before John Aloisi's side seemingly weathered the worst of a second-half storm from the visitors.
But the Reds kept on coming and eventually broke through, with substitute Ryan Kitto equalising in the 88th minute, then Johan Absalonsen scoring the winner in extra time.
Usually, it's Brisbane producing the late heroics at Suncorp Stadium. The tables turned for the Reds to open the new season with two consecutive wins.
It was a deserved result after Adelaide wrestled back control of a match that ebbed and flowed with both teams enjoying periods of dominance.
Italian veteran Maccarone started the ball rolling, smashing in his first A-League goal at the back post after he was set up by a Corey Brown cross, much to the delight of the 11,178-strong crowd.
The Roar had an array of chances to make it 2-0 but goalkeeper Paul Izzo made six important saves - snuffing out a dangerous Peter Skapetis break, another Maccarone effort and a rasping shot from substitute Brett Holman.
The 22-year-old made a huge blooper in the 48th minute when he coughed up possession near the six-yard box and was cornered by Maccarone and Nick D'Agostino. He made up for it by blocking back-to-back efforts from D'Agostino to keep Adelaide in the game.
United almost four minutes later after George Blackwood beat his marker Avram Papadopoulos and cut into the box, only for his shot to cannon off the post, into goalkeeper Michael Theo's back and then go out for a corner.
They continued to rally and were rewarded for their persistence, with Kitto making it 1-1 just minutes after coming onto the field.
Danish winger Absalonsen completed the comeback, scoring his second goal in as many games.
Reds coach Marco Kurz was delighted with the win but less impressed with his team's error rate.
"We can do a lot of things better; I think we make too many mistakes," he said.
"It's not possible to come back in every game (like this). But I hope now, with the victory, we have more confidence in our quality."
Aloisi said the Roar paid the price for not finding a second goal and killing off the game.
"It's disappointing not to win. It's disappointing to have two losses in a row," he said.
"But we can either roll over and feel sorry for ourselves or we can fight and make sure we're ready for the next game.
"I know what I'm going to do - I expect our players to fight as well."