Vince dived at the near post to meet Scott Floyd’s telling cross and ultimately seal the Pilgrims’ progress to the third round of the competition.
United had numerous opportunities to kill the tie off during the second half but progressed nonetheless to book a home clash against Guiseley on Wednesday.
Boston (4-3-3): Harry Payne; Jack McCallum, Kofi Ross, Elliot Andrew, Bailey Forth; Jack Dykes, Harry Vince, Oli Drummond (George Hobbins 55); Scott Floyd (James Gordon 83), Kemal Yenibertiz, Tinashe Chavundura (Peter Evison 71). Subs (not used): Liam Mead, Travis Portas (gk).