Van Persie believes Solskjaer must deliver not just trophies but also attractive football to win over United supporters.
He told The Sun: "Question marks over Ole will remain unless something changes, even if they win the Europa League — the standards are that high.
"It would help take some pressure off, but the club have been winning over so many years and everyone remembers the Ferguson era.
"His philosophy was about taking risks and playing entertaining football and that's what United fans expect.
"When you play there that's the pressure you have to get used to. Winning is not enough — you have to win in a nice and attractive way."
Van Persie explained: "When they won under Mourinho, somehow it didn't feel satisfying. They won it, yes, but you have to do it in a certain way. If you win ugly, I don't think it counts for some people.
"If you go back to that final against Ajax, it was a tactical final. The games before that were not great either. You were not describing United as being a couple of levels above their opponent.
"They got over the line in the final, but over the next couple of months Solskjaer needs to go that one stage further. Football is entertainment and fans want to see that. Miss a chance? That's fine, but it has to be in a way that pleases the eye.
"Yes, there are many styles to win; there is more than one way to Rome. But the system they find has to be a positive one.
"And I hope for Solskjaer he can find that style, whatever that is, and do it in a classy way.
"He has proven himself in Europe before but if you are the coach of United you always have to prove yourself. That never stops, from the first day to the last."