Everton had been fortunate to lead when a corner deflected in off Victor Lindelof's leg, though Ole Gunnar Solskjaer's side looked very different to the one that deservedly won the Manchester derby last week.
They got themselves back in the game when the 18-year-old Greenwood finished from outside the box after coming off the bench.
United missed a chance to move within two points of the top four and now sit in fifth, four points behind Chelsea. Everton, meanwhile, stay in 16th but with a new-found belief. Ferguson's management, characteristically passionate, has connected the fans with the players.
Story of the game
A fast United start saw Jesse Lingard hook a loose ball wide of the right post on his birthday. The Reds were faced with a solid defensive block of Everton, intentionally designed to frustrate.
Chances arrived for Dan James and Marcus Rashford, but United's play was slow and uninventive for the most part. Rashford's moment came after a wonderfully weighted and directed ball from Lindelof in defence. Rashford took it down well but somewhat scuffed his finish wide of the near post.
James came close moments later, curving an effort wide of the far post this time. He'd been fed through by McTominay who paused, waiting for the Welshman's run, and delivered the right pass. Pickford could have come and claimed possession but was indecisive and instead backtracked as James missed the opportunity.
United were certainly the dominant side and Everton midfielder Tom Davies took himself perilously close to a red card early on. The Englishman was booked in the fifth minute for a reckless challenge on McTominay. His late, silly challenges continued until the break.
It wasn't a miserable half for United, but they were struggling. Martial and Rashford attempted to play inside the Everton box but things just didn't quite come off. Rashford's powerful free-kick was parried by Pickford. James enjoyed a couple of quick runs past Leighton Baines, who replaced the injured Lucas Digne in the 24th minute.
In defence, Maguire's passing was inconsistent and Shaw's clearances inexplicable. One header fell at the feet of Dominic Calvert-Lewin who rasped a shot into De Gea's grateful hands. The next allowed Everton to win a corner.
And from it, came another Duncan Ferguson jump and sprint down a touchline. Despite the rain, Ferguson had removed his jacket. Down to a shirt and tie, he celebrated once again as Everton boss as a Lindelof own goal gifted Everton a lead. The corner had gone over the heads of all in the box before deflecting in off the knee of Lindelof. De Gea appeared to have been fouled by a Calvert-Lewin arm but VAR checked and said no.
United drifted into half-time and then out of it. Poor efforts from Fred and Rashford after the break were from 25 yards out and that summed things up. When Shaw stung the hands of Pickford with a fierce drive, Dan James followed it up. His effort bounced back off the face of Lingard. That also summed things up.
Mason Greenwood, 18, arrived onto the scene to add pace, energy and drive for United. James switched over to the left-wing and Greenwood took his place on the right. A left-footed Lindelof curler, from a corner, gave hope to United with 20 minutes remaining.
Were it not for the sheer excellence of Aaron Wan-Bissaka's tackling, more chances would have been offer to Everton from the lapses of concentration that United showed.
And were it not for the brilliance of Greenwood's finishing, perhaps United would never have found a way back in this game. But even at 18, he's a forward whose finishing, and hunger for goals, you can rely on.
As the ball came to him on the right of the Everton penalty area, Greenwood had a glance up and struck it low and hard through a pair of legs and into the very corner of the goal's right side. That was his third goal in a couple of games.
Everton, unlike in many games this season, bounced back. Alex Iwobi forced De Gea into action once more. Calvert-Lewis won a free-kick just outside the area.
Takeaways from the game
Mason Greenwood shows his quality again
The 18-year-old replaced Lingard and made the difference for United. They'd missed a few first half chances, relatively easy ones too. Greenwood put away a difficult one. Solskjaer's hailed his finishing in the past and won't hesitate to again after that strike.
Luke Shaw still has much to prove
The English full-back proved himself in part with a good performance in the Manchester derby. But his lack of attacking output combined with a string of mistimed or mishit defensive clearances ALMOST cost United the game on Sunday.
With Brandon Williams sitting on the United bench and very much a first team player these days, the pressure is on Shaw and he's yet to show that he deserves to be first-choice.
Inconsistent United strike again
United have now taken just nine points off bottom half sides this season. Fewer than any other team in the top half. The remaining 16 points have been earned against top half teams. They've won more points (13) against the five teams above them than against the 14 teams below them (12).
It's a woeful record. Big wins against big sides have provided brilliant days out for United's fanbase. But to finish the season in the top four, which is likelier now than ever before for Solskjaer, his team must beat the smaller sides. And they still can't break them down. They need more creativity. Perhaps Pogba's reintroduction will change that, but perhaps not.