Kastrati was recommended to the club by first-team goalkeeper Glenn Morris after a successful period with the Crawley Performance Academy.
Morris himself is heavily involved with the Crawley Performance Academy and has previously contributed to the signings of Rafiq Khaleel and Szymon Kowalczyk.
Morris said: “Florian has worked really hard since coming into training. It’s brilliant for him personally and we’re so pleased it’s finally sorted.
“Along with Szymon and Raf, it’s another fantastic achievement from all involved at the performance academy. Florian can work towards the next step in his development now. The future looks bright!”