The Coasters booked their place in the fourth round following a 2-1 win over Leyton Orient Academy.
First half goals from Stan Holder and Mason McKay proved the difference in the capital on Monday evening.
AFC Fylde - the only remaining representatives in the competition from the National League U19 Alliance - led by two goals before the hosts got a goal back in the second half.
Holder broke the deadlock after 30 minutes before McKay smashed in a second on the stroke of half-time - Devine Samuel's back-heel reply proving to be a consolation.
"We’ve made history and no-one can take that away from them," Delfouneso beamed after the win.
"Let’s enjoy the moment because they’ve deserved it.
"From the start of the day all the way to this very moment, I couldn’t be any prouder of the team, the staff and everyone involved.
"The hard work, the grind and the sacrifice that each and every one of them in that changing room today, I can’t be any prouder.
"I think you’ve seen our identity today which we abide by. It’s who we are. The desire, the passion and the heart from everyone you saw today, and we left nothing out there.
"We want to go into every game with no regrets and we want to know we gave everything, and they did.
"One of the biggest things I said from day one was about being a team and being together. I think that showed tonight."