United have won just three major honours since Sir Alex Ferguson retired in 2013 - the FA Cup, League Cup and Europa League under former managers Louis van Gaal and Jose Mourinho.
The last of those triumphs was in 2017 and although current boss Solskjaer guided United to the semi-finals of the FA Cup, Carabao Cup and Europa League last season - as well as a third-place finish in the Premier League - the club's wait for silverware continues.
United travel to Goodison Park to face Everton on Wednesday night in the last eight of the Carabao Cup, live on Sky Sports, and Solskjaer stressed the importance of winning his first trophy as United manager.
"Of course it's something we're aiming for," he said. "We want to improve every season and to improve on last year is to get to the final. Of course, when you get to a final, there's only one thing that matters and that's lifting the trophy.
"I remember winning my first trophy and towards the end of my career, I think Patrice Evra and
Nemanja Vidic's first trophy was the League Cup in 2006. It does give something to a team.
"We are desperate to get our hands on a trophy. These players are desperate to learn how to win."
'Lingard to return for Everton trip'
Solskjaer also confirmed Jesse Lingard will be included in United's squad for the trip to Goodison Park.
The 28-year-old has not played since September 30 against Brighton in the Carabao Cup and not featured in the Premier League all season.
"Jesse will be in the squad. He has been working hard in training," Solskjaer said.
"He has been unfortunate in having to isolate a few times having been in close contact with Covid, so he's been very unlucky.
"He has been working really hard and he'll definitely be involved in the squad on Wednesday."
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