Freddie Eastwood got the first after good work from James Allen, who himself made it 2-0 with a free-kick that slipped through the keeper's hands.
Daryl McMahon, Allen, and Eastwood were involved in the build up to the third, a simple side footer for Niall Ritchie after Eastwood crossed from the byline.
Birmingham grabbed a last minute consolation with a penalty.
The quarter final is likely to be a week on Saturday, probably against Tottenham.
Says Peter: "It went very well in atrocious conditions - it hasn't stopped raining all day and night, and the pitch was very heavy.
"But we dug in and played some really good stuff at times - and we scored three very good goals."
The match followed hot on the heels of the boys' excellent 7-0 win over Fulham at the weekend.
"It was a tremendous performance," Peter adds. "We started really well and played some good stuff against a top class side - and it could well have gone to double figures.
"And the pleasing thing was it followed from a horrendous journey back from Dallas when we were delayed with the tornado, so we only had one training session before the game."
With the competition really hotting up now, Peter adds: "There are some really good sides left - Tottenham, Ipswich, Derby - who have only lost one game - it really starts to get tough now.
"We've done really well though and I don't think too many sides will want to face us either."
Team: Jackson, Khan, Fletcher, Ward, Dean, Allen, Ritchie (Cole), Cascione, McMahon, Eastwood (Akinsette), Mehmet (Pearson). Unused subs: G.Sealey, Tatum.
Afterwards, Peter and Tony Carr drove to Huddersfield to see Glen Johnson play his second game for England under-16s, against Switzerland. The first match ended in a 3-1 defeat by Italy.
*Speaking of rumours linking Muzzy Izzet with West Ham, the Leicester midfielder says: "I'm very happy at Leicester but I've heard about the West Ham link and they are my favourite team.
"However I don't think I'll be allowed to leave
Filbert Street; I have no idea what sort of figure it would be but
imagine it would be something very big."