Despite being under pressure throughout the first half, Edgeworth had the chance to take the lead from the spot on 25 minutes.
Bren Abbott-Hammel slotted in the penalty to make it 1-0 for the host.
The advantage lasted less than 10 minutes as Mooy hit back for the A-League side with a superb strike.
City pushed in forward in search of a winner, but it only came in stoppage time as Mooy again proved the hero for the visitor.
A-League champion Melbourne Victory had little trouble seeing off Balmain Tigers, defeating the New South Wales State League 1 club 6-0 in Sydney.
In front of 4849 fans at Leichhardt Oval, Victory needed only 10 minutes to take the lead through striker Besart Berisha.
Just before the break, the A-League champion doubled its lead, courtesy of winger Connor Pain.
The contest was put beyond doubt just before the hour, with Pain netting his second of the night to give Victory a three-goal lead.
Despite its comfortable advantage, Kevin Muscat's side continued to push forward and Daniel Georgievski made it 4-0.
Victory added a fifth as youngster George Howard finished coolly from close range.
Melbourne's marquee midfielder Fahid Ben Khalfallah then completed the rout with a sublime effort in the closing stages.
Across town, Sydney United came back from two goals down and scored three second-half goals to lead South Hobart 3-2 going into the closing minutes.
But the Tasmanian side levelled before the 90 minutes to force extra-time.
With no goals in the extra period the contest went to penalties with Sydney United prevailing 3-1 in the shootout.
In Canberra, Sydney Olympic left it late to secure a 1-0 win over Gungahlin United.
