The Reds had enough time to recover from De Gea's third-minute error but could only level through Bruno Fernandes who has now contributed five goals in six games for the club.
It was Dominic Calvert-Lewin who benefitted from De Gea's lackadaisical clearance but he couldn't find a winner in a second-half that Everton dominated.
But for a moment, Goodison Park erupted when they thought he had. VAR ruled out the goal for offside as Gylfi Sigurdsson interfered with play while on the floor. Everton boss Carlo Ancelotti was sent off after the full-time whistle as he remonstrated with referee Chris Kavanagh for the dramatic late action.
United now sit three points behind Chelsea in fourth while Everton moved level on points with Arsenal but stay 11th.
Story of the game
By half-time, Roy Keane would be calling for David de Gea to be "lynched". That was perhaps the most extreme reaction of a moment of incredible lethargy and stupidity from the Spanish goalkeeper that gifted Everton an early lead.
As Calvert-Lewin pushed forward, racing away from the Gwladys Street End, De Gea showed an almost arrogant lack of urgency. When he eventually did try to clear, his kick out ricocheted back off Calvert-Lewin and rested in the United net. Only two minutes had passed. Look away and you could have missed it. Some sitting in Goodison Park did.
De Gea had to recover quickly from yet another mistake in a season of occasional humiliation. Within two minutes he was diving to his left to deny Calvert-Lewin an instant second. It was a good save one-on-one.
A frantic start continued when Nemanja Matic clattered the crossbar with a left-footed drive. United were opening up space in the Everton defence with some ease, their diamond formation causing trouble for the Toffees.
Shaw fizzed a powerful ball into Anthony Martial's feet but his control was poor. To make it up for it, he tried an almost-half-volley and his shot dribbled wide pathetically. Only nine minutes had gone.
Another Matic fierce drive, this time tickling the tips of the grass, saw Pickford push out into his own area. As the game lulled, United captain Maguire tried to step out of defence and push his side forward. It worked but still no finish was there to rescue the Reds from De Gea's early mistake.
An injury to Seamus Coleman disrupted the flow of the game but Djribil Sidibe, his replacement, had only been on for three minutes when Bruno Fernandes provided another piece of individual brilliance. His third goal for the club since signing in January was a swerving effort from outside the area. Jordan Pickford should have done better and tipped it wide. He, like De Gea, has had a season that brings into question his position as number one for his national side.
A frantic half concluded with a flurry of yellow cards, four in four minutes. Tom Davies was fortunate not to be sent off before the break after two infringements seemed worthy of bookings. Calvert-Lewin and Shaw were first to be booked after a coming together near the halfway line. That should have been Davies' first time after clattering into Shaw while the left-back was already down. He soon picked up that booking for a late challenge on Matic. Maguire then joined him as he halted an Everton counter-attack. Richarlison missed a diving header and the referee wrapped up proceedings.
Everton recovered over half-time after United had threatened more in the first-half. Gylfi Sigurdsson hit the post with a wonderful free-kick that De Gea had oddly left to chance.
Greenwood was too slow on the counter for United but eventually created a chance for Martial whose shot was saved. United's quick passing of the first half - which, were it not for the catastrophic De Gea error, would have seen them ahead - was absent for most of the second. Everton grew into the game and Calvert-Lewin excelled with his aerial presence and pace. Maguire covered for Lindelof time and time again.
Fred was left aggrieved when a ball that struck his thigh was given as handball and a sixth booking of the day glinted in the Sunday sun. Solskjaer brought Odion Ighalo and Juan Mata on, both coming in off good midweek performances.
Fernandes continued to be central to United's play, driving forward the counters and pressing Pickford hard. His role changed as the draw grew staler, pushing forward and playing as the third striker. A quick run and good delivery was only blocked by 35-year-old Leighton Baines who had sprinted most of the pitch to be perfectly positioned.
Pickford produced a match-defining save, after his earlier weakness, to deny Ighalo a second United goal in as many matches. The England goalkeeper raced across his goal and his left leg blocked Ighalo's close-range effort. Fernandes had already forced a save a moment earlier.
The other end turned even more dramatic. De Gea first made a stunning save but Everton followed it up well. As Sigurdsson lay on the floor in front of De Gea, a Calvert-Lewin shot trickled in at the near post. Goodison erupted. Everton sprinted to Carlo Ancelotti's touchline but VAR adjudged Sigurdsson to be interfering with play. A frantic match finished level. Ancelotti marched to the officials after full-time and was shown red.
Post-match dissection followed. Calvert-Lewin admitted he was unsure as he wheeled away in celebration but was convinced it should have stood after watching a replay. De Gea moved to save the shot and so clearly saw where it was headed before it took a deflection off Maguire. But Sigurdsson was quickly interfering with play. It was a tough decision and one taken relatively quickly by the VAR who must have been certain.
Takeaways from the match
Solskjaer continues to cause problems for opposition
Though Everton don't belong to the 'top six', a trip to Goodison remains a big game for United. Once more, Solskjaer started with a diamond midfield that Matic sat the base of. Fred and McTominay did well as they split a little wider and Fernandes enjoyed freedom in front of them all. It was another game where Solskjaer's tackles caused issues, and not for United.
Everton struggled to cope with United's set up, seemingly undecided on who should be marking who. It's not the first time Solskjaer has shifted things about game-by-game since the start of February with United reaping the rewards home and away.
Calvert-Lewin leads line for Toffees
His first-half goal may have relied on De Gea's mistake but Calvert-Lewin was Everton's main attacking threat. He did well in the air, beat Lindelof for pace and held the ball up well whether through strength or skill.
Dean Henderson's stock rises without playing a game
22-year-old Dean Henderson has had a superb season in the red and white of Sheffield United. He will have been watching his parent club, Manchester United. De Gea made the first mistake and that opens up the debate over his future again. Henderson is the ready-made replacement.
Before the first half had ended, Pickford was under the microscope too. Henderson's claim for the number one spot at United and England benefitted greatly today.