The Spain international started in the Manchester United goal ahead of England goalkeeper Dean Henderson following a strong second season under Ole Gunnar Solskjaer.
Manchester United appeared to be cruising towards a second successive Premier League win after Edinson Cavani and Bruno Fernandes got on the score-sheet in the first half.
However, Everton managed to navigate their way back into the Premier League clash after De Gea’s mistake allowed Abdoulaye Doucoure to halve the deficit in the 49th minute.
The Toffees were level two minutes later when James Rodriguez scored a clinical goal before Scott McTominay’s header restored Manchester United’s tentative lead.