Norwich, who reached the quarter-finals by winning at Everton on penalties, blew their golden opportunity of forcing extra time when Nick Davey somehow skied his spot kick way over the bar in the 79th minute.
That left Slovakian Robbie Mak’s superb 19th-minute strike for the Cup holders past outstanding England Under-17 keeper Declan Rudd as the only difference between the sides in front of an anxious 3,254 crowd.
Academy manager Cassell said: “It was a difficult game for us and I’m delighted with the result but disappointed with the performance, but credit to the opposition, they were fantastic on their special day.
“They were lifted by playing here and if the penalty had gone in it might have been a different story, although we felt we had the chances to put the game away and should have had a penalty ourselves.
“We were seen to be favourites but it can be difficult to knock that three-foot putt in. You can’t churn out the performances as you’d like week in, week out, and sometimes you have to graft and hope for that bit of luck.
“We allowed Norwich too much time, we usually play a faster tempo than that, but the pitch was a bit difficult today and maybe the occasion affected a few. They are young boys after all, not seasoned professionals.
“But we are in the semi-finals for the third time in four years, and it will be too very good teams battling it out. Arsenal are excellent. We will have to be at our very best over two legs to beat them.”
Top scorer Mak started his 18th birthday celebrations 24 hours early with a fabulous finish for his 25th goal of the season - well on target for the 30-mark that coach Alex Gibson has set his star striker.
Bratislava-born Mak, who has been with City three years, turned down the chance to join Chelsea and opted instead for City because he felt he would have more of a chance to make his mark at senior level.
He said: “Scoring goals is my job so I am very happy. That was a fantastic finish, I really liked it. It was a little bit lucky. I finished with my toe! But it was great to have a crowd, it does inspire you.”