Vaughan was 16 years and 270 days old when David Moyes brought him off the bench to score in Everton’s 4-0 win over Crystal Palace on April 10, 2005.
Dowman slashed 197 days off his record when scoring in Arsenal's win against Everton on Saturday to become the Premier League's youngest goalscorer.
Vaughan, now Everton head of academy recruitment, told Sky Sports: “It was crazy to be honest with you. My little boy loves going to the Everton games and we’d never been to an away fixture, so I’d promised I’d take him to one.
“It was only the day before, we decided to go down to Arsenal. So, we were in the stadium and I had a feeling all day.
“I’d known for a while to be honest, if I’m truthful, I know how good Max is.
“At the same time, I think I got away with one earlier, last season with Rio Ngumoha at Liverpool, both of those are two players who I thought for a while were going to break the record.
“To be in the stadium, with my little boy watching, I think he was a little bit disappointed, but for me, I couldn’t think of a better player to break it.
“I’ve watched him for a few years now. One thing you don’t want to do is put too much pressure on him.
“His life is going to change from this day forward but he’s a special talent and I expect to see massive things for him in the future.
“It’s probably an honour that it’s him who has broken it after all the players that have come before.”