The Red Devils managed 14 shots on target but only just scrapped through with a 1-0 extra-time victory.
They will now face either Wolves or Sevilla in the semi-finals in Dusseldorf on Monday night.
Speaking after the game, Solskjaer said: "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.
"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.
"It's Wolves or Sevilla, two good teams, we don't mind."