United bowed out 3-2, after extra-time, in their second qualifying round encounter against neighbours Lincoln City.
Leon Loftus, Liam Harris and Jenk Acar were the City scorers after 120 minutes of pulsating cup football.
Boston (4-3-3): Harry Payne; Jack McCallum, Kofi Ross, Daniel Moulds, Bailey Forth; Jack Dykes (George Hobbins 99), Harry Vince, Cameron Dear; Harry Limb, Kemal Yenibertiz, Tinashe Chavundura (Oli Drummond 54). Subs (not used): Conor Allen, James Gordon, Travis Portas (gk).
Referee: Brian O'Sullivan.
Attendance: 199.