The former Arsenal captain has been in sensational form for the Red Devils since moving to Old Trafford in the summer, leading the Premier League goalscoring charts with 18 strikes as United top the table five points clear of last season's champions Manchester City.
However, Meulensteen suggests the deal for Van Persie only got the go-ahead once both he and the Scottish manager were sure the squad would be able to deal with the arrival of a high-profile player from a rival side.
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"We said to them, 'What would you think if Robin van Persie comes here?' They both said the same thing. They both thought it would be outrageous if Robin came to United - and later so did all the other players.
"The reason we discussed it with Ryan and Paul is because they stand for the culture of Manchester United in the dressing room. They keep that culture and protect it.
"Robin is an absolute professional. He's a fantastic trainer and he delivers on the pitch.
"That creates a security for players. They feel nothing bad can happen to them because we have got Robin van Persie. We really missed that kind of influence last season."
The coach hailed the Netherlands striker's talents, praising his ability to find space and make things happen when the pressure is on, which rubs off on the rest of the squad.
"Robin has a phenomenal eye for free space," he continued. "I don't know another player who creates space for himself so well.
"He gives the team an invincible feeling. They think ‘We've got Van Persie. With him we are always going to score. We can always give him the ball and he will always do something magical with it.'"
And after being edged out by neighbours City on goal difference in the Premier League last year, Meulensteen states it was necessary for United to get another world-class player to push them on to the next level.
"It was time for Manchester United to have a big-name player like Robin van Persie again," he added. "In the last few seasons, we had squeezed everything out of our star players and last season we were just not good enough.
"He's a really world-class act. As one of the biggest clubs in the world we really needed a player like him.
"All Robin is focused on is success - he has a massive desire to win. I know because he could have earned so much more money elsewhere."