While Solskjaer has improved United's Premier League standing since taking charge in December 2018, he is yet to win a trophy and has been beaten in all four of the semi-finals he has reached.
That included the semi-final in the Europa League last season against eventual winners Sevilla.
United is at home for the first leg against Roma, a team they have played six times before – all in the Champions League – winning four.
One of those victories was the famous 7-1 win in the second leg of a 2007 Champions League quarter-final tie that Solskjaer played in, the spirit from which he hopes to rekindle at Old Trafford on Thursday.
"It will be a dream come true the day I can lift a trophy for this club as a manager," said Solskjaer.
"That is what we are building for. It is ifs and buts, but our ambition is to finish this season with a trophy and to celebrate. I know what that can mean for the group."
Asked if he had conducted a detailed analysis on the previous semi-final defeats, Solskjaer added: "We have looked at them and the reasons why.
"But every semi-final is a tough one against tough teams.
"Some defeats you dwell on a little bit more, but it is about putting things right and arriving with an almost fully fit squad.
"Now we are going to go all out, we play Liverpool in between but that will take care of itself.
"I would like to think [we have more belief now]. It was a strange end to last season with the mini tournament [in Germany]; the atmosphere and feeling.
"We have to go that one step further and win. We have had a great night in this tournament before [winning it in 2017]. Our focus is on ending the season with a celebration.
"As soon as we went out of the Champions League our focus was getting to a final. We feel we have been in a proper competition, we have enjoyed it.
"When you come to the business end of the season, first of all being there for the important games, having confidence and belief, but of course to have that luck and quality which is needed.
"We have done remarkably well to get to five semis. Now we want to go all the way."
United has only lost one of their last 12 games against Italian opponents in European competition (W8 D3).
The only defeat in this run came at home to Juventus in the Champions League group stage in 2018-19.
Roma, which is seventh in Serie A, has failed to win any of its last 12 away games in Europe against English clubs, a run that has factored in matches against seven different sides (D4 L8).
On the 2007 tie with Roma, Solskjaer said: "I remember both games, I started the first one. We dug out a very good result after 2-1 defeat [in the first leg].
"That night was magical. Alan Smith was excellent, we showed what United can do. We want a performance like this again!"
United has no new injuries after the 0-0 Premier League draw at Leeds, with Eric Bailly and Marcus Rashford available while Anthony Martial and Phil Jones are still out.