Cosmos Mathwasa and Harry Schofield had fired the visiting Nomads into the lead on separate occasions but Limb's 41st, 56th and 60th minute goals put the Pilgrims in command.
Tinashe Chavundura completed the scoring from the penalty spot to book a second qualifying round clash at home to Lincoln City in two weeks' time.
Boston (4-3-3): Harry Payne; Jack McCallum, Kofi Ross (Conor Allen 81), Daniel Moulds, Bailey Forth; Jack Dykes, Harry Vince, Oli Drummond (George Hobbins 54); Harry Limb (Scott Floyd 75), Kemal Yenibertiz, Tinashe Chavundura. Subs (not used): James Gordon, Travis Portas (gk).
Referee: David Avison.
Attendance: 109.