Pollock struck in the 51st minute to head home Billy Chadwick's corner delivery - in a match which contained little goalmouth action.
United needed big saves from Cameron Gregory to deny Jacob Blyth in the first half and Sam Fielding after the break - while Tom Nixon was red-carded for a second bookable offence with four minutes to play.
The Pilgrims changed shape ahead of kick-off, with York City loanee Sam Sanders handed a full debut. On-loan Birmingham City midfielder Finley Thorndike, meanwhile, was named on the bench.
Sam McLintock lifted an early effort off target, following good play from Billy Chadwick, but the save of the first half denied to Gregory, who produced a huge right hand to keep out Blyth's goal bound header from a Lucas Odunston cross.
Ex-Pilgrim George Sykes-Kenworthy produced a fine save to deny McLintock's header at the beginning of the second half, but the resultant Chadwick corner found POLLOCK, who powerfully headed home his first goal for the club.
Gregory took centre stage again when he tipped Fielding's drive on to the post, following Will Longbottom's corner.
Avenue threw plenty forward in the closing stages, but United twice went close through McLintock and Thorndike - either side of Nixon's 86th minute red card, which was the first dismissal of the Ian Culverhouse era.
Boston (3-4-1-2): Cameron Gregory; Ben Pollock, Luke Shiels, Sam Sanders (Lirak Hasani 46); Tom Nixon, Zak Mills, Tom Platt, Ethan Sephton; Sam McLintock (Finley Thorndike 72); Jordan Burrow, Billy Chadwick (Will Atkinson 88). Subs (not used): Tre Mitford, Keenan Ferguson.
Bradford (4-2-3-1): George Sykes-Kenworthy; Lucas Odunston, Mitch Lund (Simon Richman 45), Luca Havern, Reece Staunton; Jamie Spencer, Will Lancaster; Harrison Hopper (Sam Fielding 62), Brad Dockerty (Josh Hinds 72), Will Longbottom; Jacob Blyth. Subs (not used): Lewis Coulton, Daniel Benson.
Referee: Ben Tomlinson.
Attendance: 1,560 (33 visitors).
PIC: Tim Hoff.