Barker said that she did not think she could place having been boxed in with a few laps to go, and was even more pleased with the medal as her family were able to watch her.
"My positioning was far from perfect and I thought I had got boxed in by Laura Trott," she told BBC Sport. "I thought with the kick that she's got I wouldn't get out of it and then I have no idea how but I wriggled my way to the front.
"I'm so happy with that and I was so close to not medalling at all that I celebrated like I'd won it at the end.
"[The support] is absolutely incredible. Theres a whole wall of Welsh flags out there which is unreal to see. My parents are here and they never get to see me race so I'm very proud today."
Australian pair Annette Edmondson and Amy Cure won gold and silver respectively.