Dowman, 16 years and 73 days old, set the record after scoring with virtually the last kick of the match at Emirates Stadium, running more than half the length of the pitch to slot home.
While Arsenal will try to make sure the youngster keeps his feet on the ground, it was important Dowman be given room to grow, said Rooney, who scored for Everton against Arsenal in 2002 at the age of 16 years and 360 days.
"Let him enjoy it and go and express himself," the former England forward said on BBC's 'The Wayne Rooney Show'.
"Mikel Arteta, some of the senior players in the Arsenal squad, will not let him get ahead of himself. But I think you have to enjoy it as well.
"So when we've got a young lad like that coming through as a 16-year-old, he's clearly got a lot of ability and you just hope he can go and fulfil that potential."
Arsenal, who are top of the league with 70 points from 31 matches, next face Bayer Leverkusen on Tuesday in the second leg of their Champions League tie.