In a wild and wonderful contest in the Danish capital, United showed guile and grit for periods and twice looked to have claimed a crucial win before succumbing late on to a goal from 17-year-old substitute Roony Bardghji.
They were cruising midway through the first half thanks to Rasmus Hojlund's double against his hometown club - both close-range finishes from a striker knowing exactly where he needed to be and when.
But the game turned in the 42nd minute when referee Donatas Rumsas was called to the pitchside monitor to rule on a late tackle by Rashford to the ankle of Elias Jelert and responded by showing the forward a red card.
With 13 minutes added, largely as a result of an injury to United defender Jonny Evans and a medical emergency in the crowd, the home side took advantage through Mohamed Elyounoussi's finish and a Diogo Goncalves penalty - awarded for a handball by Harry Maguire.
United defied their numerical disadvantage to hold the home side at bay, before again claiming the lead through Bruno Fernandes' penalty - awarded via VAR for a handball by Lukas Lerager.
But the home side were not to be denied. Lerager made amends and drew them level with a back-post effort, before Bardghji hammered home after United failed to clear.
While United are down, they are not yet out, although they do now sit bottom of Group A and face a game at Galatasaray - where they have never won - before hosting the already-qualified Bayern Munich.
Manchester United seem determined to put themselves and their supporters through an emotional wringer this season.
You would forgive most for believing that their worst start to a campaign in six decades, culminating with back-to-back 3-0 home losses to Manchester City and Newcastle, would be the nadir.
That certainly seemed to be the case when Hojlund gave them a 2-0 lead that their purposeful and precise display fully merited. At that stage, the Red Devils were eyeing a second successive win, following Saturday's snatched victory at Fulham, and a much-strengthened shot at making the last 16 of Europe's top club competition.
Little did we know of the drama to come.
Rashford sparked it when his mis-planted foot - clumsy at best - found the ankle of Jelert and gave the officials a decision to make, which they did to United's detriment. It sparked an implosion when this still-fragile United could least afford one.
Suddenly, United's composure gave way to chaos, with Copenhagen drawing strength from a raucous and re-energised Parken Stadium to draw level. United were wobbling, the whistle for half-time timely.
Restructured and refocused by manager Erik ten Hag, the visitors were excellent for 25 minutes. Their resilience at the back was complemented by hints of counter-attacking intent that kept Copenhagen cautious, even with a man extra. It was at the end of this that they won the penalty, from which Fernandes gave them the lead again.
When the dust settles, it will be this period and the matching one in the first half that Ten Hag will likely draw on to show that his battered and bruised side are indeed making progress, albeit slowly.
Ultimately though, with such little wiggle-room following their early-season travails, results are key for United now, and once again one eluded them here.
The instinct to retreat to their own box was understandable in the circumstances, but the defending when they got there for Copenhagen's leveller and winning goal left a lot to be desired.
In contrast to Old Trafford two weeks ago, it was the Danes dancing deliriously after a late show of strength. They are now second in the group, leaving United with it all to do to keep their European campaign alive.