Zeki Amdouni’s calm late penalty moved Burnley within two points of safety with three matches remaining.
All the Clarets’ hard work appeared to be in vain when Antony broke the deadlock, but they responded well to ensure a hard-earned share of the spoils.
Burnley remain 19th but are now two points behind 17th-place Nottingham Forest, who host Manchester City on Sunday, while United stay sixth.
How the match unfolded
Although David Datro Fofana shot straight at Andre Onana inside the opening 30 seconds, United soon established their authority on proceedings with Christian Eriksen curling just wide and Fernandes rattling Arijanet Muric’s right post as they threatened the opening goal.
But the hosts had Onana to thank for keeping Burnley at bay as the first half went on. First, he tipped over Wilson Odobert’s long-range effort, before producing a magnificent reflex save to deny Foster.
Onana kept out Foster again while Alejandro Garnacho fired into the side-netting four minutes before half-time, as both sides returned to their dressing rooms still seeking the breakthrough.
Garnacho continued to carry United’s greatest threat after the break. The Argentinian winger shot narrowly wide after a swift counter, before Muric pushed away his powerful volley from a corner.
However, the visitors gifted their opponents the opening goal in the 79th minute, as Antony intercepted Sander Berge’s loose pass before racing away and slotting past Muric.
But there was to be a late twist three minutes from time, when Amdouni coolly rolled home the equaliser from the penalty spot after Onana had fouled the Burnley striker following a VAR review.
United ended their four-match winless streak when they came from behind to beat Sheffield United in midweek.
However, they have not recorded successive victories since mid-February after Amdouni’s late penalty.
Bruno Fernandes did his best to rally the hosts, becoming the first player this season to create 100 or more chances.
It was in vain however, with Erik ten Hag’s side now having failed to keep a clean sheet in six attempts.
Vincent Kompany won five of his nine Premier League visits to Old Trafford as a player (56 per cent), with only David Silva enjoying a higher success rate among opposing players with eight or more trips (75 per cent).
The Belgian was looking to become the third person to triumph here as a player and manager, after Steve Clarke and Mark Hughes.
While he could not quite achieve that, his side were more than deserving of a point following a spirited display against United.
Now with only one defeat in their last eight matches, the Clarets have made themselves difficult to beat at a crucial time in the season, and they will hope it pays dividends in their quest for another season in the top flight.
Club reports
Man Utd report | Burnley report
Match officials
Referee: John Brooks. Assistants: Lee Betts, Akil Howson. Fourth official: Simon Hooper. VAR: Peter Bankes. Assistant VAR: Mat Wilkes.
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