Rasmus Hojlund netted twice inside the first 28 minutes to send the Red Devils two goals ahead in Denmark, but the match turned when Marcus Rashford was handed a straight red card in the 42nd minute.
Copenhagen scored twice in the latter stages of the first half to level the scores, with the home side having the momentum heading into the half-time interval.
A Bruno Fernandes penalty saw Man United regain the lead in the 69th minute, but Lukas Lerager levelled the scores at 3-3 in the 83rd minute before 17-year-old Roony Bardghji struck the winner in the final exchanges.
The result has left the 20-time English champions bottom of Group A, one point behind second-placed Copenhagen, ahead of a trip to Galatasaray at the end of the month.
Man United have now lost a Champions League match in which they led by two plus goals for the first time.
Meanwhile, the defeat was the team's first after taking a two-goal lead since September 2014, when they suffered a 5-3 loss to Leicester City in the Premier League.
In addition, Man United have now been beaten in nine of their opening 17 matches of a campaign for the first time since the 1973-74 campaign.
Bardghji, who is thought to be on the radar of a number of major clubs, also became the youngest player to score against the Red Devils in the Champions League at the age of 17 years and 358 days old.
However, there was at least one positive statistic from the match, with Hojlund becoming the first Man United player to score five goals in his first four Champions League appearances for the club.
The Denmark international has also become the youngest player to net twice in an away match in the Champions League for the 20-time English champions.
The summer arrival from Atalanta BC has been in impressive form in Europe this season, but he is still waiting to hit the back of the net in the Premier League.
Erik ten Hag's side will be bidding to bounce back from the Champions League disappointment when they resume their Premier League campaign at home to Luton Town on Saturday afternoon.