An overturned penalty, two disallowed goals and efforts off the woodwork frustrated the Old Trafford giants during a goalless 90 minutes in which the Danish side looked dangerous despite failing to register a shot on target.
Fernandes would save the day from the spot in extra time, securing a 1-0 victory and a last-four date against Wolves or Sevilla back at Stadion Koln this Sunday.
Solskjaer will need a vastly-improved performance if they are to get any further than that, though, as their tired, sloppy play was compounded by Copenhagen playing with the intensity you would expect from a side in their first-ever European quarter-final.
United struggled to find a rhythm in the stifling Cologne heat but had the better of the chances, with the VAR ruling out a penalty and then Mason Greenwood’s emphatic strike in the first half.
The match was on a knife-edge as play headed into extra time, which was just five minutes old when Fernandes scored from a spot-kick won by Martial.
Copenhagen retained hope as goalkeeper Karl-Johan Johnsson produced a string of superb saves and Victor Lindelof became the latest to hit the goal frame, but there was to be no late leveller for Stale Solbakken’s dogged side.
United struggled on a night when the mercury regularly hit 32 degrees Celsius and Copenhagen offered a surprising threat.
Brandon Williams produced a big clearance as Victor Nelsson lurked at the far post ready to direct home a corner, while Fred breathed a sigh of relief after Mohamed Daramy and Jens Stage saw shots blocked following his sloppy play.
United looked to have a chance to take the lead against the run of play when Martial went down under pressure from Nelsson, moments before Fernandes hit the crossbar, but after a delay the VAR ruled out the penalty for offside.
Another decision taken by those in front of monitors would frustrate United before half-time, with Rashford putting Greenwood in for a thumping strike that went in off the far post.
But substitute Scott McTominay was soon asking the journalists behind the bench whether Greenwood was offside – a decision that would be confirmed by the VAR after review.
United probed with increased regularity when they returned from half-time and Rashford found the net in the 57th minute, only to be ruled offside when reacting quickest to a Greenwood shot hitting the woodwork.
Fernandes became the latest to hit the goal frame when trying his luck from 25 yards, with the Danish side remaining a threat as long as the match was scoreless.
Bryan Oviedo wasted a great chance to put Copenhagen into a 65th-minute lead as Aaron Wan-Bissaka threw himself in the way of the substitute’s shot following fine build-up play.
Solbakken was furious not to be awarded a penalty when former United full-back Guillermo Varela went down under pressure from Williams, with anger turning to relief when Johnsson’s superb one-handed save stopped Martial bending home.
Nelsson’s superb challenge halted Martial’s stoppage-time slalom as a helter-skelter 90 minutes came to a conclusion, with the France international denied from an acute angle shortly after the restart.
Martial wriggled through to get away another shot saved by Johnsson and had just got back to his feet when Andreas Bjelland bundled into the back of him when collecting a Juan Mata pass.
VAR cleared referee Clement Turpin’s decision and Fernandes kept his cool to lash out of the Copenhagen goalkeeper’s reach.
Mata was impressing from the bench and saw an effort saved, with Rashford then thwarted when through one-on-one – not that it mattered as the offside flag belatedly went up.
Johnsson was impressing in the Copenhagen goal and superbly denied Fernandes, before halting impressive Mata with his feet and seeing Lindelof hit a post.
Copenhagen desperately searched for a leveller in the second period but could not find a way through.