A lightning-fast start from the EDS saw us hit the front after just seven minutes when Micah Hamilton stroked home Dire Mebude's cross.
Will Dickson grabbed a brace to put us out of sight before half-time, with his goals coming either side of a red card for United's Sonny Aljofree.
Oscar Bobb tapped home our fourth with 20 minutes to go before further goals from Kian Breckin and Jaden Heskey rounded off the scoring at the Academy Stadium.
It was a stunning win for the EDS, who go back top of the Premier League 2 as a result and Barry-Murphy was pleased with how his team applied themselves to the game.
He said: " I’m delighted with the players’ performance, their attitude and their application of the game.
"When the game was in its early stages, I thought we were exceptional.
"The second half can be difficult because I think when you play 11 v 10, you expect the opponents to stand-off.
"I thought the professionalism and the way we controlled the game in the second half was very very good and I suppose you see the end result where we can score goals later in the game."
Watch the full interview with Brian Barry-Murphy via the video above.
CITY+ AND RECAST
You can watch exclusive City content on CITY+ and Recast.
Available for just £2.99 per month across all our digit platforms, a CITY+ subscription and provides supporters with unrivalled access to new and exclusive City content.
It will allow you to watch full-match replays of all Men's and Women's matches, selected broadcasts of Women's, Elite Development Squad and Academy matches and provides access to the full catalogue of City Studios documentaries.
The official Man City Recast channel offers fans flexible, pay-per-view live fixtures and on demand viewing options including match action from across the years, legend interviews, Best of Series, Man City Esports and City Studios documentaries.