The Red Devils were kept at bay for the whole 90 minutes and needed extra-time to edge past a brave FC Copenhagen side 1-0 in a sweltering heat of the RheinEnergieStadion.
Mason Greenwood and Bruno Fernandes struck the post in normal time and the Englishman had a goal ruled out by VAR on the cusp of halftime for offside.
Copenhagen also had their goalkeeper Karl-Johan Johnsson to thank as he made 13 saves, the joint-most by a goalkeeper in a single Europa League game since the competition's rebranding in 2009-10.
Speaking to the media after the game, Solskjaer admitted that Manchester United deserved to progress and lauded Johnsson for his world class saves.
"It's the third time this season we have been to a semi-final so we are delighted to get through. We deserved it, their keeper was unbelievable and we hit the post a few times,” the Norwegian coach told reporters.
"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.”
Manchester United will face the winners of Wolves and Sevilla for a place in the Europa League final on August 21, and Solskjaer insists he is ready for any of them despite admitting they are two good teams.
He added: "It's Wolves or Sevilla, two good teams, we don't mind."
