In an action-packed finale, Adelaide's Dylan McGowan was sent off late in extra time just moments after bagging a 114th-minute winner, having profited from a remarkable pass from Marcelo Carrusca.
Rhyan Grant opened the scoring with a stunning 40-metre strike on the quarter-hour mark, only for Adelaide to equalise through Pablo Sanchez from close range at Hindmarsh Stadium.
Adelaide's Craig Goodwin clipped the crossbar with a free-kick, but a youthful Sydney line-up enjoyed the better of the second half even if both goalkeepers were rarely troubled.
Last season's runners-up Perth Glory narrowly avoided an upset with a late victory over Lions FC.
Perth faced a mountain to climb after just six minutes into their visit to Lions FC when goalkeeper Jarrod Tyson - the penalty shootout hero for the Glory in the previous round - was dismissed after just six minutes.
However, substitute goalkeeper Nick Feely pulled off a save from Andre Bonotto's resultant spot-kick to maintain parity.
Despite the numerical advantage, the in-form Queenslanders, who competed in the National Soccer League (NSL) as Brisbane Lions, were unable to find a way past Feely.
The Glory avoided the lottery of penalties as Chris Harold secured an against-the-odds win with just three minutes remaining.
Heidelberg United won through to the quarter-finals for the first time, seeing off Sydney United 2-0 in a match-up between two former NSL clubs.
Central Coast Mariners-bound striker Daniel Heffernan scored after just 25 seconds for Heidelberg - a FFA Cup record.
It took until the final minutes of the match for the Bergers to make the game safe as Ken Athiu prodded home after a defensive mix-up.