United looked set to move level on points with leaders Manchester City until Dominic Calvert-Lewin snatched a point for Everton in the 95th minute at Old Trafford.
Ole Gunnar Solskjaer’s men continued their goalscoring form from a 9-0 thrashing of Southampton in midweek as Edinson Cavani’s header and a sumptuous lob from Bruno Fernandes gave them a 2-0 half-time lead.
But the defensive problems that have blighted United’s chances of a first league title in eight years were exploited by Carlo Ancelotti’s men after half-time.
“Going forward we were very good, but we need to stop conceding easy goals,” said Solskjaer, who claimed expectations of a title challenge were asking too much of a side that finished 33 points behind champions Liverpool last season.
“We’re not talking about winning titles,” added the Norwegian. “This team has come a long way.”
Two goals in seven minutes from Abdoulaye Doucoure and James Rodriguez early in the second-half brought Everton level.
Scott McTominay’s header restored the Red Devils’ lead 20 minutes from time, but they could not hold on as Calvert-Lewin slotted home after a hopeful free-kick into the box with virtually the last kick of the game.
A point aids Everton’s bid for Champions League qualification as they close to within three points of fourth-placed Liverpool, who host City today.
Arsenal’s hopes of a top-four finish are receding as they remain in 10th after a second defeat to Villa this season.
The Gunners fell behind after just 74 seconds to Ollie Watkins’s deflected shot and were unable to muster a convincing response.
Villa, on the other hand, can dream of making the Champions League as they climb to eighth, five points off fourth with a game in hand.
The closest Mikel Arteta’s men came to a response was a stunning save from former Arsenal ‘keeper Emiliano Martinez to turn Granit Xhaka’s free-kick behind.
“In the Premier League when you give a goal to the opponent - and we gave them another three opportunities to be fair, just giving them to them - and then you are not ruthless in the opponent’s box, you don’t win games,” said a frustrated Arteta.
‘Embarrassing’ Soucek red
West Ham remain fifth, but did little to aid their chances of a top-four finish with a 0-0 draw at struggling Fulham.
Scott Parker’s Fulham are still without a win in 12 games, but could easily have shaded a game of few clear-cut chances when Ademola Lookman’s free-kick flew just wide.
However, the result was overshadowed by a controversial refereeing decision by Mike Dean deep into stoppage time when West Ham’s Tomas Soucek was sent-off after a VAR review for an unintentional clash with Aleksandar Mitrovic.
“I’m actually quite embarrassed for Mike that he would have made that decision,” said West Ham boss David Moyes.
Fulham are still eight points adrift of safety after Burnley came from behind to earn a 1-1 draw against Brighton.
Johann Berg Gudmundsson’s equaliser made amends for an earlier mistake when he let Lewis Dunk’s header squeeze past him on the line for Brighton’s opener.
A six-game unbeaten run has allowed Brighton to open up a 10-point gap on the bottom three.
Newcastle also pulled clear of the drop zone with a thrilling 3-2 win at St James’ Park to ease the pressure on beleaguered boss Steve Bruce.
Southampton got off to another nightmare start as Joe Willock struck on his Newcastle debut before two goals from Miguel Almiron, either side of Takumi Minamino’s reply for the Saints, put the Magpies 3-1 up.
A fine James Ward-Prowse free-kick dragged the visitors back into it and Southampton should have taken advantage after Jeff Hendrick’s red card and Fabian Schar’s premature departure through injury - which left Newcastle down to nine as they had used all their substitutes.
But Newcastle held on to claim just their second win in 14 attempts in all competitions as they moved 10 points clear of the relegation zone.