The interview was recorded before Andrew's own BBC interview, which prompted him to stand down from public duties because of a backlash at his defence of his friendship with Epstein.
But Virginia Giuffre called on the British public to support her as she dismissed claims initially made by the prince's backers that a photograph apparently showing him with his arm around her might not be genuine.
"The people on the inside are going to keep coming up with these ridiculous excuses like his arm was elongated or the photo was doctored," she told the BBC's Panorama programme.
"I'm calling BS (bullshit) on this. He knows what happened, I know what happened. And there's only one of us telling the truth," she said.
"It's a real photo. I've given it to the FBI for their investigation."
Andrew himself doubted the authenticity of the photograph during his badly received interview, saying "nobody can prove whether or not that photograph has been doctored."
Giuffre has alleged she was trafficked to have sex with friends of Epstein when she was 17. Epstein was found dead in a New York prison in August awaiting trial on sex trafficking charges.
She described how Epstein preyed on her once he knew she came from a vulnerable background, having already suffered sexual abuse.
The financier initially lured her with an offer as a job as a masseuse, but soon began abusing her, according to Giuffre.
She accused Epstein of bringing her to London in 2001, where she was introduced to Andrew and Epstein's socialite girlfriend Ghislaine Maxwell.
The prince asked her for a dance at a nightclub, she recalled, telling Panorama he was "the most hideous dancer I've ever seen in my life.
"His sweat was like it was raining basically everywhere," she added.
"I was just like grossed out from it but I knew I had to keep him happy because that's what Jeffrey and Ghislaine would have expected from me."
The prince told the BBC he was unable to sweat at the time due to a medical condition he developed after serving as a helicopter pilot in the Falklands War in 1982.
After the nightclub, Giuffre -- then known as Roberts -- said that Maxwell told her: "I have to do for Andrew what I do for Jeffrey and that just made me sick."
She and Queen Elizabeth II's second son later had sex at Maxwell's house in central London, she alleged.
"It was disgusting," she said. "He said 'thanks' and walked out."
Andrew, 59, said he did not remember meeting Giuffre and that he "absolutely and categorically" did not have sex with her.
But an email discovered by Panorama appeared to show Andrew writing to Maxwell in 2015 with "specific questions to ask you about Virginia Roberts".
She replied "have some info -- call me when you have a moment," according to the BBC investigation.
Giuffre admitted to having a "foggy memory" over dates and places when she was asked about inconsistencies in her story.
But she added: "I implore the people in the UK to stand up beside me, to help me fight this fight, to not accept this as being OK.
"This is not some sordid sex story. This is a story of being trafficked, This is a story of abuse and this is a story of your guy's royalty."
Panorama reported that Andrew could be served with subpoenas if he returns to the US, with lawyers for five of Epstein's victims demanding that the prince testify in their court cases abut how he witnessed people being given massages.
After his BBC interview, Andrew issued a statement in which he said "I ... unequivocally regret my ill-judged association with Jeffrey Epstein" and "deeply sympathise with everyone who has been affected and wants some form of closure".
A statement from Buckingham Palace said they "emphatically denied that The Duke of York (Andrew) had any form of sexual contact or relationship with Virginia Roberts."
Panorama also interviewed another of Epstein's victims, who made further allegations against Maxwell.
Sarah Ransome told the programme that she "controlled the girls... like the madam." (AFP)