Profligacy reared its ugly head for Erik ten Hag's troops early doors, but a 79th-minute strike from Antony - his first Premier League goal of the season - would ostensibly prove sufficient for maximum points.
However, a moment of madness from Andre Onana offered Vincent Kompany's side the chance to restore parity from the spot, which they took through Zeki Amdouni, leaving the Red Devils 12 points off the Champions League spots with just 12 left to fight for.
Aston Villa need only take one more point from their final four fixtures to ensure a year without Champions League football for Ten Hag's side, while 19th-placed Burnley have cut the gap to Luton Town and Nottingham Forest to just one and two points respectively.
The Red Devils - who were booed off the field - now hit the road to face Crystal Palace in two Mondays time, while Burnley's fight against relegation continues at home to Newcastle United on May 4.
Andre Onana shines in open first half
While there was no change in the Man United starting lineup, Mason Mount was fit enough for a place on the bench, whereas Burnley made just one change to their XI, bringing David Datro Fofana back into the XI.
A showdown between two leaky backlines gave rise to an end-to-end, frenetic battle, which started in exactly that vein; a seventh-minute Arijanet Muric save from an Antony strike was the pick of a few chances inside the opening 15 minutes.
Ten Hag's troops soon assumed control of the tie, though, and the hosts came within inches of breaking the deadlock in the 19th minute, when the influential Bruno Fernandes smacked the woodwork.
However, the momentum shifted in Burnley's favour as the second half progressed, and it took a terrific reaction save from Onana to prevent Lyle Foster's header from bulging the net in the 34th minute.
The Cameroonian repeated the trick to deny his African counterpart in the 38th minute following a slick pass from Sander Berge to send Foster away, well and truly earning his keep as the game remained goalless at the break.
Onana goes from hero to zero
Onana's opposite number Muric also came up trumps just five minutes into the second period, keeping out Antony's close-range effort from an Alejandro Garnacho cutback, and Ten Hag's side continued to push for the coveted opener as the hour mark approached.
Ten Hag's peculiar decision to take off Rasmus Hojlund and Kobbie Mainoo with his side searching for a goal did not go down well with the home crowd, who were also aggrieved when a 76th-minute penalty shout was waved away; Garnacho had gone down under pressure from Dara O'Shea.
However, smiles beamed across the Theatre of Dreams just three moments later, as Antony - who had been a threat all afternoon - intercepted a poorly-placed Berge pass, raced forward and managed to slide a tame strike into the far corner while taking a tumble.
Antony's first Premier League goal of the season was just reward for his endeavours all afternoon, but the Brazilian's joy was incredibly short-lived, as Onana inexplicably clattered into Amdouni while coming to punch away a Casemiro header back to him.
Instead, Aaron Wan-Bissaka dealt with the danger, and Onana only punched Amdouni in the head - akin to the Wolverhampton Wanderers incident in August - and the Burnley striker dusted himself down to score from the spot and draw Burnley level with 87 minutes gone.
The Red Devils were booed off the pitch after failing to muster up a response in added time, and it is almost guaranteed that they will not be lining up to the Champions League anthem for 2024-25.