United's destiny to avoid the drop effectively fell out of their own hands through Saturday evening's 4-0 defeat at home to Manchester City on the back of a 2-1 win for Burnley at Watford.
Burnley's success put the Clarets level on points with Leeds but above Jesse Marsch's side due to a significantly better goal difference of minus 15 compared to United's minus 34.
Fourth-bottom Leeds remained five points ahead of third-bottom Everton but with the Toffees having two games in hand ahead of Sunday afternoon's clash against third-placed Chelsea at Goodison Park.
A draw or Blues victory would have put United's destiny back in their own hands but Frank Lampard's side recorded a 1-0 win after a horlicks from Chelsea captain Cesar Azpilicueta near his own box.
The Chelsea captain slipped as he looked to play out from the back and Richarlison picked his pocket and was then played in by Demarai Gray before slotting home a finish past 'keeper Edouard Mendy.
The Blues looked to sure to equalise when Mason Mount let fly with a powerful volley from inside the box but his shot hit both posts before bouncing out to Azpilicueta whose follow up attempt was somehow saved by Jordan Pickford.
From the subsequent corner, Pickford produced another brilliant save to deny a ferocious drive from Antonio Rudiger with his face but Everton almost bagged a second in the closing stages when Gray walloped a shot over the bar.
The Blues were given seven minutes of added time to bag an equaliser but Mateo Kovacic was denied by another strong Pickford save.
Leeds are now just two points ahead of Everton who still have a game in hand, the Toffees having five games left but Leeds just four.
Everton's goal difference is minus 20, 14 goals better than United's.
The Toffees are away at Leicester City next Sunday afternoon in a 2pm kick-off at the same time as Leeds take on Arsenal at the Emirates.
Burnley are at home to Aston Villa the previous day.
Burnley: Aston Villa (h), Tottenham (a), Aston Villa (a), Newcastle (h).
Leeds United: Arsenal (a), Chelsea (h), Brighton (h), Brentford (a).
Everton: Leicester (a), Watford (a), Brentford (h), Crystal Palace (h), Arsenal (a).