Leighton Baines equalised with a penalty in the 89th minute after a foul by Marouane Fellaini to snatch a point for the home side at Goodison Park and condemn United to a sixth draw in nine league matches.
The goal means Jose Mourinho's side have now dropped seven points from goals conceded in the last 10 minutes of league games this season - more than any other team in England's top flight.
7 - Manchester United have dropped seven points from goals conceded in the last 10 minutes of PL games this season, a league high. Sloppy.— OptaJoe (@OptaJoe) December 4, 2016
Juan Camilo Zuniga and Troy Deeney scored in the 83rd and 95th minutes to give Watford a 3-1 win after the match was poised at 1-1 at Vicarage Road, while Joe Allen and Olivier Giroud scored respective late equalisers at Old Trafford to force 1-1 draws for Stoke and Arsenal.
The draw means United have closed the gap to fourth-placed Manchester City to nine points, although they have won only twice in the league since August.