Ryan Pratt's 31st minute opener was soon cancelled out — and the Lambs went on to book a third qualifying round home clash against Alfreton Town.
Bobby Robinson drew Tamworth level and Kyle Hayden struck twice to seal United's fate.
Boston (4-2-3-1): Seth Greeves; Marshall Young (Arkadiusz Michalczyk 75), Taylor Gray, Jack Withers, Jack Brown (Jack Henton 42); Callum Rodger, Jamie Gaukroger; Leighton Boughen, Jenson Bark, Xander Crashley; Ryan Pratt (Harrison Holland 67). Subs (not used): Jenson Ball, Oliver Swaby (gk).
Referee: Adam Clenaghan.
Attendance: 151.