United face a two-legged quarter-final against Barcelona in April after staging a dramatic comeback from a 2-0 first-leg deficit to knock out Paris Saint-Germain on away goals in the last 16.
Solskjaer scored the dramatic winner in the 1999 final as United struck twice in injury-time to beat Bayern Munich 2-1 in Barcelona and become champions of Europe for the second time in their history.
United, having won the trophy in 1968, lifted it again in 2008 by defeating Chelsea on penalties in Moscow but have not reached the final since 2011, when they were beaten 3-1 by Barcelona.
Solskjaer has acknowledged that his team benefited from a significant decision in their favour to beat PSG, with Marcus Rashford scoring the decisive goal from a penalty awarded for handball with the help of VAR in the dying minutes.
The caretaker manager recognises that United had to come through similarly tight situations to win the Champions League 20 years ago, having to come back from two goals down in the away leg to defeat Juventus in the semi-final before their dramatic late recovery against Bayern.
"I think we've shown against PSG that on any given day we can beat a top team," said Solskjaer, who has turned United's fortunes around since he took over from the sacked Jose Mourinho in December.
"Then again, there are so many variables and these games will be decided by margins.
"And of course with that VAR decision, even though for me it's a penalty, you can still be unlucky and not get it in the last minute and then we'd be out and no one would really remember the PSG performance.
"So yes, we can go all the way but you've got to be lucky and good along the way."
- Treble winners -
This year is the 20th anniversary of United's treble-winning season, when they won the Premier League and FA Cup as well as the Champions League, with difficult moments on the way to all three trophies.
"To win trophies, you've got to win at least one or two really tough games," said Solskjaer. "They're like defining moments.
"So comparisons? Well we're not challenging to win the league this year but we are challenging for the other two trophies."
Solskjaer is aware of the comparisons between United's current run and that enjoyed by the treble winners of 1999.
It has even been suggested to him that they are fated to be successful under him this season, as it is 20 years since the treble, and he used to wear the number 20 as a player at Old Trafford.
However, he argued that hard work is more important than historical coincidence in the pursuit of trophies.
"There have been a couple of comments about 'You used to wear the number 20' and blah blah blah, but I believe that you get what you deserve in life, in sports.
"If you put the work in, I believe, if you work hard, show the right attitude, if you go in every single day and do the right things, you can maximise your chances."