The Pilgrims arguably should have taken all three points against Micky Mellon’s Latics, with former loanee Tom Donaghy producing good saves to deny Jimmy Knowles and Keaton Ward, while Jai Rowe had a header controversially ruled out for offside.
Oldham did come within a whisker of nicking the win when Mike Fondop rattled the crossbar with a stoppage time free-kick.
Graham Coughlan had opted to name a completely unchanged squad for Oldham's first visit to Boston since an FA Cup tie in 1952.
Oldham's Tom Conlon crashed an early free-kick off target, while Jacob Hazel slammed an effort wide, via Charlie Raglan's block.
Hazel flicked a subsequent effort over the bar from close-range, while Donaghy dived to his left to keep out a goalbound drive from Knowles.
Oldham created a big chance four minutes into the second half, but Fondop slammed a near post drive wide after meeting Joe Pritchard's low cross from the right.
United made a treble change on the 70-minute mark, with Ward, Adam Marriott and James Gale all entering the fray.
Eight minutes later, the Pilgrims thought they had taken a decisive step towards three points, but Tim Walker's offside flag curtailed the celebrations after Rowe had headed home Dylan Hill's lofted cross, despite at least three defenders appearing to play the United man onside.
Marriott and Ward combined for the latter to test Donaghy in the closing stages, before Fondop came within centimetres of snatching an undeserved win for the visitors deep into stoppage time with a free-kick that bounced to safety off the crossbar.
Boston (3-5-2): Cameron Gregory; Connor Teale, Zak Mills, Brad Nicholson; Jai Rowe, Frankie Maguire (Keaton Ward 70), Jordan Richards, Dylan Hill, Cameron Green; Jimmy Knowles (Adam Marriott 70), Jacob Hazel (James Gale 70). Subs (not used): Tony Weston, Olly Green, Jacob Scott, Jake Lovelace (gk).
Oldham (4-2-3-1): Tom Donaghy; Reagan Ogle, Shaun Hobson, Charlie Raglan, Jake Leake; Tom Pett, Tom Conlon (Jordan Rossiter 55); Joe Pritchard, Matt Worthington (Billy Water 73), Jeserun Uchegbulam (Vimal Yoganathan 79); Mike Fondop. Subs (not used): Mark Kitching, Jake Caprice, Oliver Hammond, Jacob Dennis (gk).
Referee: Scott Simpson.
Attendance: 1,695 (273 visitors).
PIC: Russell Dossett.