Bruno Fernandes missed a 93rd-minute penalty as Aston Villa won at Old Trafford for the first time in 12 years in a pulsating match.
Kortney Hause had put Villa ahead with a powerful header two minutes from the end of normal time before the defender handled inside the box to present United with a lifeline.
However, Fernandes blazed over high over the bar in front of the Stretford End as United's unbeaten start to the Premier League season came to an end.
Having suffered an early Carabao Cup exit at home to West Ham, United have lost their past two games without scoring.
Villa missed two big chances before a dramatic finale.
After Matt Targett blazed over inside the six-yard area, Ollie Watkins could only fire at David de Gea after the United keeper had passed straight to the Villa forward following Harry Maguire's backpass.
In a thrilling game, Ezri Konsa also headed another chance over after beating Aaron Wan-Bissaka in the air, while Watkins was denied by De Gea at full stretch in the second half.
At the other end, Emiliano Martinez kept out Maguire's header and Paul Pogba nodded wide when the France midfielder should have scored.
Penalty woe for Solskjaer
Ole Gunnar Solskjaer used his pre-match news conference to voice concerns about United's lack of penalties in recent weeks.
The Red Devils boss thought his side were denied "two stonewall penalties" in a 2-1 win at West Ham last Sunday, while he also felt they should have had a spot-kick in the Carabao Cup defeat by the Hammers on Wednesday.
When substitute Edinson Cavani's header struck Hause's arm deep into stoppage time, referee Mike Dean immediately pointed to the spot.
Fernandes, however, ballooned his attempt high over Martinez's crossbar leaving Villa to celebrate a famous win.
It capped a bitterly frustrating day for United and Solskjaer, who suddenly find themselves under pressure before Villarreal's Champions League visit to Old Trafford on Wednesday.
Solskjaer is expected to deliver trophies this season after a busy - and expensive - summer in the transfer market.
With one avenue closed in the EFL Cup, they need to bounce back and beat Villarreal after losing their first group game against Young Boys.
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