A deflected strike from Anthony Gordon gave the hosts victory, a defeat interim manager Ralf Rangnick also admitted could see the Red Devils struggle to finish in the Premier League's top-six.
Even though the visitors started brightly, and had 68 percent possession, the Toffees were largely untroubled in the contest and Frank Lampard's men secured a huge boost in their quest to avoid relegation.
Rangnick's side were slow on the ball, and rarely got in behind the Everton defence.
Because of that, they succumbed to defeat, and have now won just once in their last seven games in all competitions.
Champions League football is looking more and more unlikely and De Gea echoed those thoughts at full time.
"It is a disgrace from us, we should be winning this game," he said to BT Sport.
"We don't score but we don't even create proper chances to score. I don't know what to say, to be honest.
"We knew before today they were struggling and how difficult it was going to be.
"We are not good enough, that is for sure. It will be difficult to be in the top four.
"They played Wednesday [Everton's 3-2 defeat to Burnley] and they were tired but they had more desire than us, that is not acceptable. Very sad to lose today."
Rangnick also spoke about his side's chances of qualifying for the Champions League next season, and said even finishing in the top-six could be a stretch for the out-of-form Red Devils.
"Not a good one," when asked about the impact of the result.
"When you don't win your own games, and drop points, it will be difficult to finish in the top-four, maybe even in the top-six.
"We had a good start, the first 25 minutes we were dominating the game, not always making and taking the right decisions, but what we had intended to play, had played in the first 25 minutes, and [we] should have scored.
"They defended with the whole team, they had almost all their ten outfield players around their own box.
"And in order to break that wall down you need to have more players in the box, and bring crosses into the box, whenever we did that in the two or three moments, we [created] some big chances, and this should have happened more often."