After impressing as interim manager following the sacking of Jose Mourinho, ex-United forward Solskjaer oversaw a dismal end to the season as they missed out on a top-four finish.
But Van Persie said: "I think it's the perfect match, it's just they're having a bad spell."
He added: "Back in the day it was normal for a manager to get time. Nowadays if you lose six games, you're out. Is that the solution?
"Give someone time, especially if it's a kid from the club."
United dismissed Mourinho on 18 December and planned to make a temporary appointment until the summer before filling the managerial vacancy permanently.
A day later, 46-year-old Solskjaer was named caretaker manager and United went on a 15-game unbeaten run in domestic competition, including a 100% record in his first eight matches.
The Norwegian oversaw a dramatic victory over Paris Saint-Germain in the Champions League last 16 and, despite then suffering two defeats, he was given the job full time on 28 March.
United finished the campaign with six losses and two wins from their final 10 fixtures.
"It's funny because when he started everyone was so positive, they were winning and they made a miracle by beating PSG," said Van Persie, who helped United to the Premier League title in 2013
and scored 58 goals in 105 games for them between 2012-15.
"Everyone was screaming like, 'Give the guy a permanent deal'. And then since that permanent deal the luck went away a little bit; they've been losing more games than they want to.
"But he got a three-year deal and everyone should look at the bigger picture. His presence, the way he presents the club, is the proper way in my opinion. He's positive, he wants to achieve big things with the club and that is the only way forward.
"He's a man of the club, he played there for more than 10 years, he won the Champions League - actually his goal got them the Champions League - so he has a big part over the years at the club.
"He's been a coach of the reserves, he is Manchester United. Similar, in a different way, to Ferguson. He was Manchester United and Solskjaer is Manchester United as well.
"Everyone has that at a certain time. He had a fantastic first period - now the last 12 games have been tough. OK, but next season they have to pick themselves up and go again."
Van Persie joined United under Sir alex Ferguson