Mooy bagged an impressive second-half hat-trick, and had a hand in another goal as City overran the home team.
A FFA Cup season-high 11,372 spectators crammed into the Olympic Village in Melbourne's northern suburbs hoping to see EURO 2004 winner Katsouranis inspire the former National Soccer League club to an unlikely win.
However, Melbourne City took a lead within two minutes as a smart passing movement started by Mooy ended with striker Bruno Fornaroli converting from close range.
City dominated periods of the opening half, yet it took a miraculous diving save from Tando Velaphi to deny Kaine Shepperd an equaliser, with the move started by Katsouranis featuring in a one-off guest appearance for the Bergers.
After the break, however, it was all about Mooy. The in-form Socceroos midfielder doubled the lead midway through the second half with a smart low finish from outside the penalty area, and he repeated the feat a few minutes later.
Mooy tapped in for his hat-trick in the 84th minute to become the highest scorer in this season's competition, before Fornaroli added salt to the wounds with his second and a late fifth for his team.
In the west, Perth Glory prevailed 4-2 on penalties over Western Sydney Wanderers.
It was the visitor that went ahead just prior to half time via Spanish signing Dimas but Glory hit back 14 minutes into the second-half with Nebojsa Marinkovic scoring the equaliser.
The score remained level at 1-1 after 90 minutes and the extra 30 minutes and penalties were required to separate the teams.
Perth kept its nerve the best in the shootout, with Federico Piovaccari and Scott Jamieson missing their spot kicks for the Wanderers before Glory captain Richard Garcia stepped up to slot home the winner.
The semi-final draw sees an all Melbourne affair with Hume City taking on Melbourne Victory and Perth Glory hosting Melbourne City.
