Manchester United are now through to the last four and will wait to discover if their opponents will be Wolves or Sevilla.
Manchester United were pushed all the way to extra-time by a stubborn Copenhagen side who were led by their inspired goalkeeper, Johnsson.
The Red Devils were denied a penalty and two goals due to offside calls, rattled the woodwork and registered 14 shots on target, but they eventually broke through via a penalty, after Martial was hauled down in the box, allowing Fernandes to thump home in the first half of extra-time.
Substitute Juan Mata played Anthony Martial through for the second time in quick succession and goalkeeper Karl-Johan Johnsson was equal to him again.
This time, however, Mata played the ball back in and Martial was sent tumbling to the floor by a clumsy challenge from Andreas Bjelland.
French referee Clement Turpin awarded United’s 21st penalty of this marathon season and Fernandes stepped up to convert in the 95th minute.
They will have to be much better on Sunday, whichever side they face. United have been flying since the restart but they have also looked rather ordinary at times and this was one such occasion.
Yes, they hit the woodwork several times last night and had two goals disallowed. But they also had to rely on some desperate last-ditch defending against a Copenhagen side who caused then more problems than they can have expected.
This felt like a European quarter-final in name only. A two-legged tie between United and Copenhagen in Manchester and the Danish capital would be boisterous affairs if nothing else.
Instead, there were no fans in the bars of Cologne, one of four German venues chosen to host the final stages of this competition in tournament format for the first time. No flags, no songs echoing around the city as they normally would on match day.
The atmosphere inside the 50,000-seater arena was even flatter as United played behind closed doors in Europe for the third game in a row. It should feel more natural by now, but it really doesn’t.
Having made changes in both games against LASK Linz in the last 16, Solskjaer reverted to a stronger line-up with Sergio Romero, Eric Bailly and Fred the only players who would not usually appear in his current first XI.
With just one defeat in their last 21 Europa League matches, United certainly looked confident beforehand with substitutes Victor Lindelof and Andreas Pereira relaxed enough to break off from their warm-up and play air guitar to the Red Hot Chilli Peppers on the PA system.
United struggled to click into gear going forward and Bruno Fernandes’ free kick, fired well over the bar after Jens Stage had fouled Mason Greenwood, rather summed up their performance.