Braintree had netted 25 goals in their opening four league games but were blunted by an organised United side.
Scott Floyd scored Boston's other goal in a match played on one of Boston Town's outside pitches, with the main pitch at the DWB Stadium deemed unplayable.
Boston (4-3-3): Harry Payne; James Gordon, Kofi Ross, Daniel Moulds, Bailey Forth; jack Dykes, Harry Vince, Dan Dougill; Tinashe Chavundura, Kemal Yenibertiz, Scott Floyd. Subs (ALL used): Conor Allen, Ollie Drummond, George Hobbins, Travis Portas (gk).