Leighton Baines scored from the penalty spot in the 89th minute of the game for the toffees to once again see United draw a game which they should have won.
Zlatan Ibrahimovic put United in front with a precise lob just before half-time but once again they had to settle for a point which could prove costly later in the season.
Rojo lucky to not see Red
The game began at a very high tempo with the home team really getting into the faces of the United players early on, without managing to create any meaningful chances.
The home side were aggrieved after 15 minutes when a two-footed challenge from Marcos Rojo on Idrissa Gueye only resulted in a yellow card when it probably should have been a red card.
This got the home fans riled up and from there, the game became a very scrappy one as both sets of players just couldn't really string a number of passes together.
Ibrahimovic opened the scoring after a Stekelenburg mistake
This will until just before half-time when United took the lead through a fantastic lob by Ibrahimovic. Anthony Martial provided the through pass for the goal but Martin Stekelenburg made the error by coming right out of his goal when Ibrahimovic was being tracked by his marker.
The lobbed shot took an eternity to go in though as it firstly hit the bar and then the post before finally trickling over the line and with the help of goal-line technology, the goal was awarded.
Therefore the visitors went in at half-time with huge relief knowing that they really hadn't played that well at all but found themselves in front in a game that lacked quality for most of the half.
Hosts much better in the second half
The toffees came out much more positive at the start of the second half and almost got themselves back on level terms when Kevin Mirallas burst his way through the United defence and forced David de Gea into a fantastic save.
After that chance, United started to go forward to try and get the second goal and they almost got it when a cross from Henrikh Mkhitaryan found its way to Ander Herrera at the back post, but he saw his shot come back off the bar.
All of a sudden after that the United players decided to sit back allow the hosts to take control of the game and De Gea was forced into making more saves as it was all back to wall stuff from Jose Mourinho's men late on.
Baines equalised late on for the hosts after Fellaini costs visitors
They almost held out for the win but Marouane Fellaini, who had just came onto the pitch as a substitute, fouled Gueye in the box and referee Michael Oliver awarded the penalty.
Baines, who saw his last penalty against De Gea saved, didn't miss this time as his powerful low shot went into the net to send the Goodison Park faithful into overdrive.
Both teams pushed after that to try and get a winner but ran out of time as once again United drew from a winning position in the last few minutes of a game.