Smith's 82nd minute header from a Liam Agnew cross settled the encounter in United's favour as the hosts dominated for long spells against their winless visitors.
The Pilgrims were indebted to keeper Dion-Curtis Henry and stand-in full-back Jason St Juste for preserving their lead in the final minute, but there was no doubting that United were worthy winners overall.
United boss Dennis Greene had opted for a radical overhaul of personnel ahead of kick-off, with Marcus Marshall and St Juste pressed into service at full-back, while Liam Adams and Cameron Johnson were handed first starts of the season.
Both sides made a tentative start, boasting a combined record of one win in 16 matches ahead of kick-off.
Adams’ curling effort was clutched by Joe Cracknell in the Bradford goal, while the visitors threatened with ex-Blackpool defender Matt Hill heading wide from Nicky Wroe’s deep corner.
Smith had a trio of chances before half-time to break the deadlock, heading narrowly over the bar following Marshall’s free-kick, glancing another header over the top from Agnew’s corner and then firing wide after Agnew’s cross had looped into his path.
Lewis Hilliard and Herve Pepe-Ngoma were introduced at half-time and Hilliard was soon bravely denied by a terrific Andy McWilliams block.
Jack McMillan had an audacious effort tipped over by Bradford keeper Cracknell while Smith was denied in a melee following a Boston corner.
Avenue created two good chances with Ben McKenna heading over from Leon Osborne’s cross while Chris Sharp glanced a header wide from a McKenna centre.
United took the lead with eight minutes to play when St Juste and Agnew combined with a short corner and the latter’s cross was powered home by SMITH.
There was still time for Henry to spectacularly keep out Evan Gumbs’ header from a Wroe corner before St Juste bravely denied substitute Liam Dickinson as the Pilgrims held on for a crucial victory.
Boston (4-1-3-2): Dion-Curtis Henry; Marcus Marshall, Joe Maguire, Joe Robinson, Jason St Juste; Jack McMillan (Callum Chippendale 75); Liam Adams (Herve Pepe-Ngoma 46), Liam Agnew, Cameron Johnson (Lewis Hilliard 46); Jay Rollins, Gregg Smith. Subs (not used): Ollie Price, Lamin Colley.
Bradford (4-2-3-1): Joe Cracknell; Ryan Toulson, Evan Gumbs, Matt Hill, Andy McWilliams; Nicky Wroe, Joe Kearns (Michael Monaghan 85); Ben McKenna, Sanchez Payne, Tom Davie (Leon Osborne 60); Chris Sharp (Liam Dickinson 79). Subs (not used): Harry Coates, Nicky Boshell.
Referee: Garreth Rhodes.
Attendance: 905 (18 visitors).