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.
