Jake Thompson's ninth minute goal had the Pilgrims in charge of their own destiny - but a spirited Aylestone Park comeback threatened to level proceedings and they finally got the goal they deserved in the second minute of stoppage time through Yassine Haffane.
Despite successful conversions from Freddie Shinn and captain Oliver Houlston, visiting keeper Ben Cluley saved brilliantly to deny Tyreece Keating and Josh Panton, as the away side converted all four spot-kicks to book a third qualifying round tie against Bedford Town.
Boston: Zach Marsh, Reiss Wheatley-Crane, Tyreece Keating, Elod Nadasdi, Oliver Houlston, Josh Panton, Afonso Nobre (Sam Finch 69), Jake Thompson (Denis Matei 83), Stephen Pereira (Freddie Shinn 46), Daniel Bridges (Archie Bradford 83), Oliver Callaby (Lake Taylor-Peak 60). Subs (not used): Aron Ealham, Joe Douce (gk).
Referee: Rob Holland.
Attendance: 158.