Marcus Rashford was sent off in the first half and United then conceded twice in the final ten minutes despite retaking the lead when down to ten men.
A fortnight on from injecting hope into a faltering continent campaign by narrowly beating the Danish champions, Parken witnessed a scarcely believable comeback that leaves the Red Devils bottom of Group A.
Rasmus Hojlund's brace had put United in control against his former club, but Rashford's red card sparked an incredible turnaround as Copenhagen's pre-match tifo reading "your theatre of nightmares" proved prescient.
This first half was as dramatic as it was action-packed. Hojlund's third minute opener at the end of a fine team move was followed by several breaks in play, before the Copenhagen native scored his second and was denied a hat-trick.
Things soon spectacularly unravelled for United. Rashford was sent off for catching Elias Jelert after a VAR review, with Mohamed Elyounoussi scoring before Diogo Goncalves levelled from the spot after a Harry Maguire handball.
A wild first half was followed by a pulsating second period. Skipper Bruno Fernandes' penalty put United back ahead, only for Lukas Lerager to level again and substitute Bardghji to seal an incredible Copenhagen victory.
For a while this looked like it would a rare straightforward evening for the Red Devils.
Excellent play from back to front had them ahead inside three minutes, with Rashford controlling a diagonal ball from Fernandes and playing on to Aaron Wan-Bissaka.
The right-back did well and slipped in Scott McTominay, whose first-time drive across goal was scored by Hojlund at the far post.
United looked sharp, controlled and composed during a stop-start period.
There was a halt for a pitch invader carrying a Palestine flag and another as Raphael Varane replaced the injured Jonny Evans while a medical emergency was dealt with behind by a white screen by the pitch.
There was applause as the fan left after treatment and another when it was announced they were awake and doing well considering the circumstances.
United went further ahead in the 28th minute.
Fernandes cut out a Copenhagen header and drove across the halfway line, before playing in Alejandro Garnacho. Kamil Grabara got down to to save the low effort, but Hojlund was perfectly placed for a simple tap-in.
It would have been a hat-trick and game over a minute later had Grabara not reacted brilliantly to tip over the 20-year-old's fizzing strike.
Hojlund dragged another effort across the face of goal as half-time approached as United ignored the hosts' whistles as Jelert stayed down in pain.
The stadium screens said a VAR check into Rashford's challenge on the full-back was over, but referee Robertas Valikonis was soon trotting over to the monitor.
The Lithuanian official reviewed replays and brandished a straight red card to the perplexed forward for catching Jelert - a 42nd-minute decision that sent United into a tailspin.
The visitors were threatening with new intensity and Goncalves cut back for former Southampton man Elyounoussi to sweep home three minutes later.
Copenhagen smelt blood and levelled nine minutes into 13 of added time.
Maguire was adjudged to have handled and Valikonis pointed to the spot, with Goncalves sending Andre Onana the wrong way.
Sofyan Amrabat replaced Christian Eriksen at the break as United looked to regain control, with Diogo Dalot denied either side of Denis Vavro and Viktor Claesson opportunities.
Goncalves tried his luck from distance as Copenhagen looked to capitalise on their man advantage, only for the VAR's intervention to give United a chance to go back ahead.
The ball struck Lerager's arm as he looked to clear and, upon review, a penalty was awarded.
Fernandes kept his cool under intense Copenhagen pressure and beat Grabara, celebrating in the corner as objects were thrown his way in the 69th minute.
The United skipper refused to take a corner as more rained down from the stands as the clock wound down.
Ten Hag's side had the ball in the net again but a foul had already been called and Copenhagen equalised in the 83rd minute.
Patient play ended in Rasmus Falk floating a cross to the far post, where Lerager all too easily got behind Dalot to direct in at the far post.
Copenhagen kept pushing and a deep cross was only cleared as far as Bardghji, who swivelled and struck a famous winner in the 87th minute.