Six changes for United from the win over LASK led to a recall for Aaron Wan-Bissaka, Paul Pogba, Anthony Martial, Bruno Fernandes, Mason Greenwood and Marcus Rashford.
A big opening for the Danes saw Pep Biel surge down the right and delivered a ball to Mohamed Daramy, who in turn couldn't get the better of Sergio Romero.
Misfortunate circumstances then followed the Danish side after Nicolai Boilesen twisted his hamstring three minutes later, with Swedish international Pierre Bengtsson brought on as his replacement.
The clearest opportunity went United's way after 20 minutes as Fernandes brushed the top of the crossbar with a thumping close-range header from Martial's flick-on.
Copenhagen's resilience backfired after Greenwood ran onto Rashford's flick to fire in off the far post in stoppage time, but the Video Assistant Referee disallowed ruled out the goal after Greenwood was said to have been offside.
Paul Pogba looked to end his side's frustrations after combining with Fernandes' cutback in the area ten minutes after the halftime break, but his first-time striker went over the top.
Some activity from the United bench saw Nemanja Matic and Victor Lindelof enter for the final 20 minutes of regulation time whilst frustrations threatened to boil over.
Extra-time was to determine a winner after a score-less 90 minutes, as United went ahead from the penalty spot on 95 minutes through Fernandes after Martial was brought down in the box.
Karl-Johan Johnsson stood between the Red Devils and a second goal afterwards, parrying Juan Mata and Fernandes whilst Lindelof rattled the post late as United join Inter Milan in the semi-finals.