The former United striker was handed a three-year contract on Thursday after leading the club to 14 wins from his 19 matches in charge since becoming caretaker manager in December.
Facing the media for the first time as the new full-time United manager, he said: “You know when people say ‘it’s their dream job’ – that’s more true to me than anyone. This has been my ultimate dream.
“Maybe a naive dream but I’ve always had that in my mind. I’m so honoured and privileged to have this fantastic responsibility to lead us forward.”
Solskjaer was asked if he had his own version of Sir Alex Ferguson’s famous vow to ‘knock Liverpool off their perch’ after he arrived at United, but he said: “I’m just going to be myself, as I’ve always been.
“I can’t wait to get on to the challenge of improving this great bunch of players. It’s a squad full of potential. I’ve worked with them for three months and the improvement is there to be seen.
“We know there’s so much more to come from them. The attitude has been fantastic, but we’ve got a way to go. I’m sure we will be successful.”
United were enduring their worst start to a season for 28 years prior to Solskjaer’s arrival from Norwegian side Molde, and sat 11 points off the top four in the Premier League.
However, United have won more points than any other Premier League side since the 46-year-old took charge and are just two points off the Champions League places with eight games to go.
Solskjaer also memorably led United to victory over PSG in the Champions League last 16, thanks to Marcus Rashford’s late penalty in the Parc des Princes, and is now preparing them for a quarter-final clash with Barcelona.
While they are underdogs against the Spanish champions, Solskjaer believes they can go all the way and lift the Champions League in Madrid in May.
He said: “I know the traditions of the club, the history of the club. Of course I want to be successful and lift trophies.
“We’ve got a chance. It’s going to be a tough one, a mountain to climb – but we’ve climbed a few mountains before.”