Dowman made his debut in Saturday's 5-0 win against Leeds, coming on as a second-half substitute and winning a penalty, which was converted by Viktor Gyokeres.
Rooney, who played and scored for Everton as a 16 year-old, said on his new podcast: "To play at that age in the Premier League is crazy. I know I played when I was young, but I think he's like 150 days younger than I was or more.
"It's just refreshing to see a young lad get his opportunity, and, even though he's playing for Arsenal, every time he cuts inside you go 'shoot'. You want him to put it in the top corner.
"If he gets it and scores, it's great, but I was thinking if he misses that's the difference in being an absolute success story and... all of sudden you get everyone, not everyone, but online people having a go at him," the former Manchester United striker added.
"So I was really pleased he didn't, but it was such an impressive debut."
Instant fame
For Rooney, the key now for Dowman and his family is not to get carried away with his instant fame.
"For Max and his family and friends, I'm sure they're all living in a dream and a fairy tale and it probably hasn't hit home yet," Rooney said. "But you can see his future is so bright, he is going to have a massive future.
"Everyone you speak to in football, the same name keeps coming back: Max Dowman.
"It's difficult for anyone. I think the first thing you've got to do is just stay around your family and close friends.
"Listen to them because there are going to be people out there who are trying to be your friend or trying to get in touch with you - some for good reasons, some for not so good reasons.
"You just have to make sure the people who are closer to you, you stay around them and listen to their advice rather than outsiders."
Enjoy it
And on what he can advise the teen, Rooney says it's all about enjoying the moment.
"Enjoy it. I'm sure we're going to see so much of him, so much of his talent and goals and assists, whatever," he said.
"Alan Stubbs always said to me years ago, 'make sure you enjoy it because it goes very quickly'.
"When you're young, you don't think that or you sometimes forget that, but enjoy it and make the most of it."