Rees Longville-Daley netted the winning goal in the 90th minute with the match seemingly destined for penalties.
The Pilgrims struggled to break down a stubborn Tulips' defence but did rattle the woodwork through Jay Rollins in the second half.
Boston (4-2-3-1): Christian Dibble; Kalern Thomas (Cameron Johnson 79), Joe Maguire, Herve Pepe-Ngoma, Courtney Wildin (Joe Burgess 55); Grant Roberts (Waide Fairhurst 55), Lewis Hilliard; Jay Rollins, Marcus Marshall, Jason St Juste (Callum Chippendale 66); Lamin Colley. Subs (not used): Josh Green, Liam Adams, Michael Emery (gk).
Referee: Martin Chester.
Attendance: 428.