Janujaz has enjoyed a breakthrough start to the campaign at Old Trafford, which continued last weekend as his impact from the bench helped United earn a 3-2 victory over Stoke City.
Van Persie believes that the 18-year-old is a special player who can become a huge star for United, insisting that his potential became clear during a productive pre-season.
He told The Mirror: "He's a great player and I'm a great fan of his. I thought earlier in pre-season that he has something special.
"Over the years I've seen so many young players and I can honestly say that of all the young players I've seen he is right up there in the top one or two.
"He's a special player. He will give Manchester United a great bunch of goals and great games."