United manager Sir Alex Ferguson highlighted the veteran midfielder's 61st-minute introduction as the game-changing moment and former Arsenal striker Van Persie wholeheartedly agreed.
"Scholes is a world-class player," Van Persie said. "You could see when we were 2-1 down, when he came on every single pass he got spot on from 30 or 40 metres. Short passes as well and because of him we won the game.
"When he came on, he changed the game so he is my man of the match."
He was also "relieved" to have made amends for his lackadaisical penalty by netting a hat-trick as Manchester United staged an incredible late comeback at Southampton.
The Holland international looked to have cost his new team a point after wasting a chance to level in the second half with a cheeky attempt that was saved by Kelvin Davis.
Van Persie managed to avoid Sir Alex Ferguson's wrath by scoring twice late on to earn United a 3-2 victory, but knows his mistake almost cost them.
"I am so relieved," he said. "When I missed the penalty, I was so gutted.
"I try to keep a certain standard in my game and that is not part of my ID, that standard.
"I will play football well, respect it and something went wrong there. I wanted to hit it high and in the corner like I always do so something went wrong.
"It is just me to blame and I am so relieved that we got the three points.
"I had a couple of chances. I missed a couple too, which is part of the game. I am very relieved to make up for my mistake."
The defeat was harsh on Southampton, whose position at the bottom of the table with zero points belies their performance against United and at Manchester City on the opening day.
In both matches Nigel Adkins' side gave up a 2-1 advantage to lose 3-2 and midfielder Morgan Schneiderlin knows Saints need to learn how to close games out.
"We gave everything for 87 minutes and in the final five minutes we lost the game," he said.
"That is what we have to learn, to win a game. It is the second time we have been 2-1 up and lost 3-2 at the end of the game so it is like all the effort we have put in is for nothing.
"But, at the end of the day, we learned a lot from this game and we're going to work very hard on the training ground on the bad things.
"Man United won the game and we're very down, but there is a lot of positive to take out of this game.
"You could say that we missed six points against the two best sides in England.
"I think we did very well against them and now we need to focus on the next game.
"We have a little break where we can focus on everything and hopefully against Arsenal we can be on top again, but hopefully win the game."