The save acted as a springboard for Brisbane to take the lead, as Thomas Broich's inswinging free kick evaded everyone in the box and curled inside the far post in the 20th minute.
However, Bruce Djite levelled with a thunderous strike 13 minutes later for Adelaide United.
Nine minutes after the restart, James Donachie turned Cirio's cross past his own keeper to hand Adelaide the lead for the first time in the match.
Matters got worse for the defending champions in the 72nd minute, when Matt McKay was shown a straight red card.
Adelaide should have put the game out of sight late on when Djite rattled the bar with a header, but they still held to secure all three points.