Jake Wright’s seventh goal of the season handed the Pilgrims the lead on the stroke of half-time.
Simon Richman equalised for the hosts with 20 minutes to play and Jacob Blyth’s flying header won it 12 minutes from time.
The match took a while to get going at the Horsfall Stadium, but United soon strung some decent moves together, with Wright and Tom Nixon both firing efforts off target following Scott Pollock throughballs.
The Pilgrims took a 45th minute lead when Scott Pollock threaded McLintock through to cross from the right - and WRIGHT swept the ball home from close-range to score for the second successive season in Bradford.
Ex-Pilgrim Jordan Preston should have equalised in stoppage time, but crashed an effort into the side-netting after getting in behind.
The second half was fairly nondescript until Avenue fashioned their 70th minute leveller, with Lucas Odunston teeing up RICHMAN to finish well on the angle.
United's afternoon took a decisive turn for the worse eight minutes later when Reece Staunton was afforded too much time to cross from the left and substitute BLYTH ghosted in between three defenders to dispatch a free header past a stranded Sam Long.
Bradford (4-2-3-1): George Sykes-Kenworthy; Lucas Odunston (Harrison Hopper 81), Will Lancaster, Gianluca Havern, Reece Staunton; Jamie Spencer, Sam Fielding; Brad Dockerty (Jacob Blyth 69), Simon Richman, Mitch Lund (Will Longbottom 57); Jordan Preston. Subs (not used): Tom Angell, Adam Nowakowski.
Boston (4-2-3-1): Sam Long; Keenan Ferguson, Jo Cummings, Ben Pollock, Tom Nixon (Alex Brown 71); Tom Platt, Lebrun Mbeka (Will Atkinson 83); Jordon Crawford, Scott Pollock, Sam McLintock; Jake Wright (Jaanai Gordon 73). Subs (not used): Tom Solanke, Rhys Davies (gk).
Referee: Richard Holmes.
Attendance: 471.
PIC: Ken Fox.