Jamie Gaukroger's long-range effort handed the Pilgrims a 49th minute lead but Stamford equalised a minute later and won it with a quarter-of-an-hour to play.
United twice rattled the crossbar and both Taylor Gray and Gaukroger were both denied by last-ditch goalline clearances.
Boston (4-2-3-1): Jake Frestle; Nathan Stevens, Callum Rodger, Daniel Fisher (Brogan Horniman 65), Owen Sears (Taylor Gray 46); Marshall Young, Lewis Bontoft (Finlay Armond 46); Jamie Gaukroger, Mido Turki (Kyle Tate 75), Jenson Bark; Will Jackson. Sub (not used): Harrison Holland.