The result, in conjunction with Perth Glory's expulsion from the finals for salary cap breaches, has allowed the Roar to rise from the ashes.
Serbian striker Andrija Kaluderovic got the visitors off the mark in the 26th minute before Reds keeper Eugene Galekovic netted an own goal.
Defender Jack Hingert added further salt to Adelaide's wounds with his maiden A-League goal early in the second half, but the the hosts refused to give up.
Craig Goodwin reduced the deficit in the 71st minute and Sergio Cirio converted a fortuitous penalty in the dying seconds after Daniel Bowles was red carded deep into stoppage-time, but it was too little, too late.
Elsewhere, Central Coast Mariners beat bottom club Western Sydney Wanderers 1-0 thanks to Joshua Rose's second-minute goal.
