The Pilgrims were 3-1 victors over Basford United at Greenwich Avenue to book a trip to Mansfield Town in round one.
Josh Lee’s 80th minute strike for the hosts was nothing more than a consolation — as United reached the competition proper for the fifth time in eight seasons.
Boston (4-3-3): Jamie Bradley; Daniel Worthington, Billy Hodgkiss, Joe Cain, Jack McCallum; Miles Lynch, Liam Moran, Corey Newman; Dominic Goddard (Jack Dykes 74), Louis Saebe, Bradley Easey. Subs (not used): Ollie Price, Steven Dovey, Daniel Moulds, Alex Dempster.