The signing of the season, Bruno Fernandes, came to United's rescue with a rocket of a penalty in extra-time.
United had two goals ruled offside in regular time and dominated for large periods of the game, however, Copenhagen defended resiliently with goalkeeper, Karl-Johan Johnsson, playing a blinder.
Story of the game
United came into the game as favourites and started well in the opening minutes. Fred put out an amazing diagonal pass for Fernandes but he couldn't make the most of it.
FC Copenhagen settled into the game as it progressed and came close to scoring on a couple of occasions. They threatened Sergio Romero's goal a quarter into the match but some brilliant defending by Harry Maguire and Paul Pogba denied the Danish team.
In the 20th Minute, a penalty was awarded to United but after a VAR checks it was overturned for an offside.
Another major incident took place on the brink of half-time when Mason Greenwood's brilliant finish from a Marcus Rashford pass resulted in a goal but again VAR ruled it out for offside.
Both sides experienced good periods of play in the first half with United slightly dominant but failed to take advantage of their opportunities.
United started the second-half on front foot when Greenwood's shot hit the post and Rashford put it in the back of the net only to be ruled offside.
FC Copenhagen soon got a chance on the counter but Romero came up with a save.
United's defenders Maguire and Eric Bailly were cautioned in quick succession and soon Bailly was substituted for Victor Lindelof.
As the match headed towards the final ten minutes, former red Guillermo Varela went down in the penalty area from a Brandon Williams challenge but the referee waved play on.
In the stoppage time Martial nearly clinched a winner for United only for a last ditch tackle to deny him.
The second half came to an end in a stalemate and headed for extra-time.
One change after full-time as Juan Mata was introduced to the game for Greenwood. Arguably he was the difference for United as he made an instant impact.
Early in the first half of extra time Martial won the penalty for Man Utd after he was brought down in the box. Fernandes kept his cool to break the deadlock and United took a 1-0 lead.
United had numerous chances to double their lead but were denied by a determined goalkeeper who played the match of his life and kept his side within a fighting chance.
Unfortunately for the Danish side they failed to equalise and United came out victorious on a hot and sticky night in Germany.
Semi-finals up next for United and Solskjaer
United have now advanced to the semi-finals of the European competition and will find out their opponent tomorrow evening when Wolves play Sevilla. This will be the third semi-final of the season for United, where they will look to pickup a trophy for the first time this season.