The Portuguese playmaker made no mistake from the spot in the 95nd minute to break the deadlock after Anthony Martial had been fouled inside the box.
United had been frustrated all night by a resilient Copenhagen side during the first 90 minutes, mainly in part due to two scoring attempts that had been fended off by the woodwork.
The sticks aided the Danish side again five minutes later when Fernandes' dipping effort beat keeper Karl-Johan Johnsson but bounced off the left post.
However, Januaray signing Fernandes would not be denied in extra minutes as United took their 21st penalty (converting 17) of the season across all competitions.
"It's the third time this season we have been to a semifinal so we are delighted to get through," United manager Ole Gunnar Solskjaer said after the match. "We deserved it, their keeper was unbelievable and we hit the post a few times.
"It could have been one of those nights that you end up with a shootout. We had to block, defend well and they made it hard for us. They were well organised with a gameplan, I knew they would be hard to break down so we had to be patient. We knew we would get chances, our attackers will always create chances.
Fernandes now leads the Europa League scorers list with seven goals, and is the joint-top scorer for United alongside Martial on 11 goals (all competitions).
"He knows keepers will wait for him to do the jump. He practices both of them [run-ups] and he practices both sides, so he's got them sorted," Solskjaer added on Fernandes' penalty-takinga abilities."
The scoreline soured Johnsson's excellent night in the net for Copenhagen, whose 13 saves were the joint most in a single game since the Europa League started in 2009.