Top of the Pops audience member Sylvia Edwards recalled how the late presenter put his hand up her skirt while delivering a link on live TV.
Edwards - who was 19 at the time of the incident - told The Sun: "I felt his fingers go towards my bottom. It was disgusting."
In a recorded clip of the show, Savile delivers a link while sat among female audience members. Facing the camera, he says: "I tell you something, a fella could get used to this, as it 'appens, he really could get used to it."
Edwards can then be seen shrieking and squirming in her seat as she tries to move away from the giggling former DJ.
"Jimmy Savile appeared in the middle of us and the camera panned around. Then I felt his hand go up my skirt. I leapt off my chair in shock. I was so surprised I cried out and didn't know how to deal with it," said Edwards.
"He just laughed and carried on mauling me while talking to the camera. I panicked and tried to move away from him but it was so crowded I couldn't escape. When I tried to sit down his hand was still there and went for my bottom again.
"I felt so embarrassed and ashamed because it was live on TV and all my friends and family were watching.
"The worst thing was that he was so casual when he did it. He was committing a sexual assault live on the BBC and no-one gave a damn."
Edwards claimed that as the show went off air, she ran straight to a BBC floor manager and told him what Savile had done.
"There was a man standing next to a camera with headphones on who seemed to be running the show, so I went up to him and told him Jimmy Savile had just put his hand up my skirt," she said.
"He was very cross and told me to get lost. He said it was just Jimmy messing about and I was being stupid. Then he said he was busy and moved me out of the way because I was blocking a camera shot."
She continued: "I was a naive girl then - I'd never slept with anyone - and was shocked. I didn't know where to go or who to speak to after getting rejected like that.
"I tried to shut it out of my mind as best as I could. But looking back, I wish I'd had the courage to take my complaint further. The evidence on the video is clear, and if I could have got him convicted, I might have saved other young people from much worse."
The incident - which took place at BBC Television Centre on November 25, 1976 and was likely to have been watched by up to 20m viewers at home - is said to be the first video evidence of Savile's alleged catalogue of abuse.
Edwards said that she was left with mental scars following the events and admitted she blamed herself for Savile's actions for years.
"What he did to me was minor compared to some. But it affects me to this day," she told the paper. "I have never felt comfortable about being touched.
"All these years I've never said anything because I thought no-one would listen after what the BBC man said. But I have to stop blaming myself because, at the end of the day, he did that to me on live TV and they let him."
She added: "You can hear Savile on the video saying, 'A fella could get used to this'. The tragedy for his victims is, that's just what he did."
Today (October 26) Savile's family released a statement expressing their sympathy for the late entertainer's actions.
"Our hearts go out to them and we offer them our sympathy and understanding in their anguish," they said.
" Our feelings are in turmoil as we await the next turn of events."
Photo gallery - Jimmy Savile in pictures:
Copyright: flickr/BBC Pictures