A-League Wrap: Adelaide, Perth win on the road
The champions looked to be on course for three points at the Suncorp Stadium when German midfielder Thomas Broich put them ahead with an expertly taken free-kick after 20 minutes.
However, Bruce Djite brought Adelaide level with a brilliant strike just after the half-hour mark, as he lashed a long-range effort past Jamie Young.
And the visitors edged into the lead early in the second half courtesy of an own goal by James Donachie.
A slick Adelaide counter-attack ended with Donachie bundling Cirio's superb cross past his own keeper at the far post, and the away team held on to secure a valuable win.
In Sunday's other A-League fixture, Jamie MacLaren struck late to give Perth Glory a 2-1 win at Wellington Phoenix.
Andy Keogh put Perth in front with a debut goal after 32 minutes, but Michael McGlinchey appeared to have earned the hosts a draw when he equalised 17 minutes from time.
But MacLaren's fine finish in the 86th minute winner stole all three points for Perth.