The Toffees produced an incredible comeback to beat Crystal Palace 3-2 at Goodison Park having trailed 2-0 at the interval and Frank Lampard's side are now four points clear of the drop zone.
The battle to avoid finishing in the division's final relegation place and joining Watford and Norwich City in being relegated is now a two-horse race between Leeds and Burnley heading into Sunday's final day.
But United's far inferior goal difference means that the Whites will be consigned to the drop if Burnley beat Newcastle United at Turf Moor, regardless of how Leeds fare at Brentford.
Jesse Marsch's side are level on points with the Clarets but their goal difference is minus 38 compared to Burnley's minus 18 so Leeds need to better Burnley's result against the Magpies.
Burnley received a big boost ahead of their Villa Park assignment as centre-back James Tarkowski returned to the side following a hamstring injury.
The setback forced the defender out of Sunday's 1-0 loss at Tottenham Hotspur but the 29-year-old returned to the XI to replace Matt Lowton as the only change to the Clarets side.Steven Gerrard made four changes to his Villa team and opted to drop Philippe Coutinho and Danny Ings to the bench along with Marvelous Nakamba
The injured Ezri Konsa missed out altogether as Calum Chambers, Jacob Ramsey, Emi Buendía and 18-year-old Carney Chukwuemeka all came into the side.
There was controversy in the sixth minute as Ashley Barnes caught Tyrone Mings with an elbow in the face.
VAR reviewed the incident and viewed the challenge as reckless but not violent.
Nick Pope then produced two smart saves to deny John McGinn a Villa opener but Burnley threatened too and the Clarets were awarded a penalty on the stroke of half time as Emi Buendia tripped Maxwel Cornet in the box.
Barnes stepped up to take the spot kick and sent Emi Martinez the wrong way to put the visitors 1-0 up at the break.
Villa hit back to level just three minutes after the restart as Buendia volleyed home McGinn's raking cross.
But Gerrard's side then survived Burnley's shouts for a penalty as Lucas Digne handballed in a crowded area from a Clarets corner.
Both sides then had chances to bag a winner and Clarets substitute Wout Weghorst squandered a sitter from close range which was kept out by a flying Tyrone Mings block.
At the other end, Pope produced another outstanding save to keep out a Bertrand Traore header.
The Clarets then finished the game with ten men as substitute Matt Lowton was shown a straight red card for a late tackle on Calum Chambers but the visitors held on for a potentially priceless point.
Over at Goodison Park, Everton were denied an opener as Richarlison's free-kick came back off the crossbar and Palace went ahead in the 23rd minute through Jean-Philippe Mateta.
The Eagles then doubled their advantage 15 minutes later through Jordan Ayew but the Toffees pulled a goal back nine minutes into the second half through Michael Keane.
Richarlison then drew the Toffees level with 15 minutes left and Dominic Calvert-Lewin sealed an amazing comeback by netting an 85th-minute winner.