The Toffees - with Goodison Park a traditionally far from happy hunting ground for the Reds - got off to a dream start only two minutes in.
David de Gea inexplicably dawdled on the ball and his eventual clearance cannoned off the onrushing Dominic Calvert - Lewin and looped in to put United's unbeaten run under threat.
United equalised after half an hour when Bruno Fernandes' 20 yarder bounced in front of Jordan Pickford, who should have done better with a shot that was straight at him.
Fernandes and substitute Odion Ighalo came within inches of a winner as Pickford redeemed himself but then came a major talking point that, almost inevitably, involved VAR.
Aaron Wan Bissaka's challenge left Gylfi Sigurdsson sprawled on the ground as Calvert - Lewin's shot struck the otherwise impressive Harry Maguire and wrong-footed De Gea. Goodison Park erupted in celebration, but the jubilation was short lived as on-field referee Chris Kavanagh went to the video technology monitor.
Replays showed that, although Sigurdsson didn't touch the ball, he moved out the way to allow the shot to come in and was lying in an offside position directly in front of De Gea. His movement affected the trajectory of the ball and he in De Gea's line of vision, so therefore the goal was correctly disallowed.
That led to the usually affable Ancelotti angrily confronting Mr Kavanagh and he was shown a red card after making his feelings known to the referee.
Although you can feel a degree of sympathy for Everton, it would have been harsh on United as, in truth, a point apiece was a fair result. That in itself represents progress as we were thumped 4-0 on the same ground last season, a performance that was among the worst in recent memory.
Our number one had plenty of time to clear following Maguire's sensible and careful backpass, but the dithering and dallying De Gea allowed Calvert - Lewin to close him down, with fatal consequences. All the Everton striker had to do was stick his leg out as the ball ricocheted off him and in, the striker's 15th goal of the season. No player has made more errors directly leading to a goal since the start of last season than De Gea's seven.
To his credit, the United keeper quickly atoned for his error when he tipped wide from the same player moments later and also saved well from the scourge of Old Trafford Sigurdsson when he tried his luck from distance.
United could have been 2-0 down but finally got a foothold as Nemanja Matic struck the bar and Mason Greenwood headed wide before the equaliser came shortly after the half hour mark.
Matic found Fernandes 20 yards from goal and for once the January signing's shot lacked power.
At a good height for the England keeper, Pickford should have saved it but instead the ball bounced under his flailing arms for Fernandes third goal in as many games and his first from open play. It was a mistake not as bad as De Gea's but still goes down as a major error in front of the watching England manager Gareth Southgate.
Matic, outstanding throughout, had a shot beaten out by Pickford but back came Everton as Richarlison flicked a header inches wide at the other end in an increasingly exciting end to end encounter.
United were the better side in the first half, but Ancelotti's revved up side came out with a spring in step after the interval.
Fred was ludicrously booked for handball when the ball instead struck his knee, and it would have been hugely controversial had Everton scored from the resultant free-kick. They nearly did, but fortunately Sigurdsson's superb effort thumped back off the post with De Gea beaten.
Anthony Martial had an effort deflected over after neat build up between Greenwood and Fernandes, but De Gea then continued his duel with Calvert - Lewin as he saved superbly with his feet.
Former Red Michael Keane cleared under pressure from Fernandes and a flurry of Everton corners came to nothing as our defence - marshalled superbly by the immense Maguire - stood strong.
As the clock ticked down, United launced one last attack and almost stole it at the death. Fernandes was inevitably at the heart of things as Pickford clawed away his goalbound effort, then somehow got back up to block Ighalo's snapshot on the rebound.
That seemed to be the last of the drama until De Gea made a brilliant point blank save from Sigurdsson and then, in the ensuing melee, came the VAR intervention in the final seconds.
Overall team performance: 6/10. Good first half, disappointing second half
United Man of the Match: Nemanja Matic. AWB, Maguire and Bruno were all outstanding, but Matic was simply immense.