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It was a wild night at Parken which looked like it was going to be anything but early on when Rasmus Hojlund gave United an early 2-0 lead.
The visitors were in complete control when Marcus Rashford was shown a straight red shortly before half-time for an accidental, but high boot on Elias Jelert.
By the break the scores were level as Mohamed Elyounoussi and then Diogo Gonçalves from the penalty spot levelled things up.
The second half turned out to be just as frantic, with Bruno Fernandes putting United back ahead from the spot, but Copenhagen roared back ahead late on.
Lukas Lerager prodded home on 83 minutes, before substitute Roony Bardghji fired home the winner just four minutes later.
Manchester United are now bottom of their group stage with just two games to play, at Galatasaray and at home against Bayern Munich. It is not over as just one point separates second and fourth place, but there is a lot of work to do.
Things could not have started any better for the visitors, with Hojlund scoring just three minutes in, continuing his brilliant record in the competition.
There was fine work on the right as Aaron Wan-Bissaka played a reverse pass to an onrushing Scott McTominay in the inside right channel, he flashed the ball across goal and Hojlund just had to tap it into an open goal.
There was concern for United 10 minutes later as Jonny Evans limped off injured, replaced by Raphael Varane, but they were 2-0 ahead shortly afterwards.
Hojlund doubled his tally at the end of a swift counter-attack, Bruno Fernandes played in Alejandro Garnacho on the left, who burst in on goal, saw his shot saved and the Dane was there to poke home. In doing so he became the quickest Manchester United player ever to reach five Champions League goals, in just four games.
It was all going swimmingly, and then it wasn’t, when Rashford was shown a straight red card for a misplaced boot on Elias Jelert.
That was on 42 minutes and by half-time the game was level. A cross from the right to the back post was volleyed back across goal by Goncalves for Elyounoussi to calmly finish from close range on 45 minutes.
After some lengthy delays in the first half – due to Evans’ injury and an emergency in the crowd – there was plenty of stoppage time and seven minutes in, United conceded a penalty.
The ball bobbled around in the area and seemed to hit Varane’s hand before it definitely hit Harry Maguire’s and the referee pointed to the spot, with Goncalves firing home.
It was a remarkably quiet start to the second half and not until the 69th minute that the fifth goal arrived, again from the penalty spot.
Lukas Lerager was harshly, but probably correctly, judged to have handballed a deep cross and, after a VAR referral it was another penalty, which Fernandes blasted into the back of the net.
Lerager got his revenge 15 minutes later though, sneaking in at the back post to get on the end of an in-swinging cross from the left from Rasmus Falk, beating Diogo Dalot to the ball and prodding past Andre Onana.
17-year-old substitute Roony Bardghji was the hero as he fired home the winner on 87 minutes, pouncing on a loose ball in the box which United failed to clear.
MORE : Marcus Rashford red card slammed by Paul Scholes and Owen Hargreaves
MORE : Paul Scholes reveals ‘worrying’ theory on why Raphael Varane has been dropped
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