The substitute turned the game after coming on just after the hour as the League Two side scored twice late on to dump Villa out.
Campbell, who is on loan from Aberdeen, started the comeback when he robbed Leander Dendoncker and was brought down by the midfielder in the box with five minutes left.
Dendoncker was sent off and Jamie Reid levelled from the spot, with Campbell firing in a dramatic winner two minutes later.
“The best part about it is my mum and dad came down from Scotland and my girlfriend is here so for them to experience that moment with me, it’s a really good feeling,” the 21-year-old told the Comet after earning a fourth round trip to Stoke.
“I’m buzzing with that. It’s an incredible moment and obviously the best I’ve had in my career so far.
“The celebrations were amazing, what a group we have. The gaffer has managed to put together a group of really top guys and the feeling was shared at the end.
“Everyone earned that, everyone deserved the win and I’m just proud of everyone.
“I heard in the back of my ear the gaffer (Steve Evans) screaming at me to go and get the corner.
“When I looked up I saw I had a lot of space and it was just about focusing on getting a good connection on it and if it goes in, it goes in. I’ve struck it really well and I’m buzzing that it has gone in.
“The lads dug deep incredibly for the first 60 minutes, they were absolutely brilliant. Villa are very good, very technical and passed it really well.
“The gaffer just wanted me to go on and try and make some passes, turn them when we can and get us up the pitch.”
It was an embarrassing exit for Villa, who had taken a first-half lead after Morgan Sanson’s first goal for the club.
“I’m upset, disappointed and sorry for our supporters. We have lost a very good opportunity to do something in the FA Cup,” said boss Unai Emery.
“I’m very shocked. But it can happen when you have good chances to score but don’t take them.
“They had chances as well. Our mistake was a big mistake and they scored.
“It is a process. In other matches we have not been clinical. We have to improve, we have to learn, we have to try and find a solution.”