The Pilgrims trailed to Lewis Knight's 36th minute free-kick but turned the tables after the break through Ben Davies' penalty and Max Wright's deflected effort within a three-minute spell.
United looked well set for victory until Ben McKenna sailed home his own deflected effort four minutes into time added on.
Pilgrims’ boss Craig Elliott had made three changes to his side ahead of kick-off, with Mansfield Town loanee Lewis Gibbens handed his first start. Jay Rollins started a game for the first time since March.
George Willis, named man-of-the-match on Boxing Day, continued his fine form by denying Oli Johnson after five minutes, with Ryan Qualter thwarting Alex Hurst on the follow-up.
Willis made another save 10 minutes later to deny Johnson with his legs.
Avenue needed a solid block from Shane Killock to ward off the danger against Rollins, before Mark Ross blocked at close-range to deny the fit-again Gavin Allott.
Bradford took the lead in freak circumstances nine minutes before the break when KNIGHT saw his audacious 35-yard free-kick deceive Willis and creep into the top corner of the net.
Knight almost added a second with a nonchalant chip towards the break but the Pilgrims were almost level before half-time, with ex-Pilgrim Jake Beesley blocking on the line to deny Allott.
United levelled six minutes after the break when Killock was adjudged to have fouled Nicky Walker in the box and DAVIES lifted home his first goal in Boston colours.
The Pilgrims completed the turnaround three minutes later when WRIGHT blazed an effort into the bottom corner, via a deflection off Killock, following industrious play from Andi Thanoj initially and then Walker.
United rattled the crossbar through Wright, following Rollins’ cutback from the left, as Boston’s chance to edge two goals to the good passed them by.
Willis made two phenomenal saves to thwart former team-mate Beesley as the Pilgrims moved closer to victory but Avenue finally grabbed their equaliser deep into stoppage time when McKENNA saw his 20-yard drive take a big deflection and nestle into the corner.
Boston (4-4-2): George Willis; Ben Davies, Ryan Qualter, Lewis Gibbens, Ashley Jackson; Max Wright, Brad Abbott, Andi Thanoj, Nicky Walker (Jonathan Wafula 83); Gavin Allott (Spencer Harris 89), Jay Rollins (Jordan Slew 68). Subs (not used): Tai-Reece Chisholm, Danny Rowe.
Bradford (4-2-3-1): Steven Drench; Mark Ross, Gianluca Havern, Shane Killock, Danny Lowe; Nicky Wroe (Nicky Clee 76), Conor Branson; Lewis Knight, Alex Hurst (Ben McKenna 81), Oli Johnson (Adam Nowakowski 71); Jake Beesley. Subs (not used): Ryan Toulson, Jamie Spencer.
Referee: David McNamara.
Attendance: 1,061 (56 visitors).