Solskjaer is only the fifth United manager to lead the club to a European semi-final.
Copenhagen's resilient defence frustrated United at times but Bruno Fernandes came to the rescue as he converted from the penalty spot and sent United to their 17th European semi-final.
Solskjaer's verdict on the performance
As the match progressed, United looked increasingly likely to grab the winner, which they did in the 96th minute. Solskjaer told BT Sport: "We deserved to win tonight, their keeper was fantastic, in fact unbelievable. It could have been one of those nights that you end up with a penalty shootout, we had to block, defend well and they made it hard for us.
"They were well organised and had a game plan, they had a decent chance in the first half but a good recovery from the lads. I knew they were going to be hard to break down, we had to be patient and earn chances. With Anthony, Marcus, Mason and Paul we will always create chances."
When discussing Fernandes' different approach to the penalty, Solskjaer said:
"He knows that keepers will wait for him to do the jump. He practices both (techniques) of them and both sides so he's got them sorted."
Solskjaer also commented in his post-match press conference on the impact Juan Mata had as he came on from the bench : "When he came on he is so composed, calm, clever, has quick feet in tight areas to pick a pass and calms us down at times.
"He is a great influence is the dressing room and I commend him for his professionalism, he is one of the top assist makers in the Europa League, he has done really well for us."
Preparation for semi-final
Unlike regular tournaments, Solskjaer will not be able to scout out his semi-final opponents as the team will be forced to stay in their 'bubble' due to COVID-19 restrictions. Solskjaer told the media how his side can prepare for Sunday:
"Will have to watch (Sevilla vs Wolves) on tele, the facilities are not the same we are used to but that's no excuse, every team has the same conditions. We are used to the best in the world for recovery and we have to make the best of the conditions with the hot and humid."
Fixture congestion has been a recent issue since football restarted but Solskjaer looks forward to his side's preparations for the semi-final: "We have six days, there is not many times we have six days to prepare for a game," Solskjaer said.
For United and Solskjaer their job is done, however, another test is on the horizon as they will face either Wolverhampton Wanderers or Sevilla FC in the semi-final on Sunday night.