James Ball's header in the sixth minute of added time rescued a point for the visitors who had seemingly been dead and buried following Gregg Smith and Jay Rollins goals either side of half-time.
Matty Warburton had cut the deficit for Stockport within five minutes of falling two goals behind and it was left to Ball's late show to deny Karl Hawley and his United side a second successive win.
Hawley had been forced into a change ahead of kick-off, with Jan Yeomans replacing the injured James Clifton.
Home keeper George Willis repelled an effort from Kallum Mantack in the game’s first meaningful opening after 20 minutes.
Ten minutes later it was the Pilgrims who went in front when SMITH powered home a close-range header following Sam Muggleton long throw-in.
The Pilgrims doubled their lead nine minutes after the break when keeper Ben Hinchliffe failed to get enough purchase on a punch from another Muggleton long throw and ROLLINS netted at the second attempt at the back post.
WARBURTON drew the Hatters back into contention with a sublime 25-yard free-kick in the 59th minute which curled around Willis’ wall and into the bottom corner of the net.
United had chances to kill the contest with Jordan Keane’s deflected effort testing Hinchliffe and Smith’s thumping drive flying wide.
Smith headed wide from a Muggleton cross and then crashed an effort over the bar following a centre from substitute Tyrell Waite.
The Pilgrims’ hearts were broken when, despite only five additional minutes being added, BALL got up highest to power home Scott Duxbury’s cross in the sixth additional minute.
Boston (4-3-3): George Willis; Jan Yeomans, Brad McGowan, Brad Beatson, Sam Muggleton; Harry Vince (Jack Broadhead 85), Jamie McGuire, Jordan Keane; Jay Rollins (Tyrell Waite 89), Gregg Smith, Ashley Hemmings. Subs (not used): Kabongo Tshimanga, Morgan Penfold, Liam Adams.
Stockport (4-2-3-1): Ben Hinchliffe; Dan Cowan, Michael Clarke, Stephen O’Halloran (Adam Thomas 77), Scott Duxbury; James Ball, Gary Stopforth; Kallum Mantack (Sam Walker 82), Bohan Dixon (Darren Stephenson 77), Matty Warburton; Jason Oswell. Subs (not used): Chris Smalley, Ian Ormson (gk).
Referee: Sam Mulhall.
Attendance: 1,211 (174 visitors).