Calvert-Lewin thought he had won the game late on but VAR ruled out the goal in controversial fashion.
Everton took the lead inside the first three minutes when De Gea took too much time on the ball and his clearance hit Calvert-Lewin and hit the back of the net.
De Gea made up for his error when he pulled off a fine save to deny Richarlison minutes later.
In a lively opening to the game, Nemanja Matic smashed the crossbar from long range and also had a shot saved by Pickford.
United levelled after 31 minutes when Pickford misjudged the flight of Fernandes' shot and the ball flew into the bottom corner.
The hosts almost regained the lead just before the hour mark but Gylfi Sigurdsson's free kick struck the post.
With a minute remaining, Pickford made a fantastic double save to deny Fernandes and Odion Ighalo to ensure the draw.
The hosts thought they had scored a late winner through Calvert-Lewin but VAR controversially ruled it out with Sigurdsson in an offiside position.
Everton boss Carlo Ancelotti was shown a red card after the match after confronting the officials at full time.
Ancelotti said he would not be speaking to Pickford about the mistake and said the ups and downs were part of the goalkeeper's fate.
"He did well. I don't want to judge individual performances. He tried to do his best - sometimes it was good and sometimes he could do better, but this is the life of the goalkeeper," said the Italian.