Pam Ward, from Edinburgh, said their family will never get over the allegations of sex abuse carried out by the former Jim'll Fix It and Top of the Pops presenter.
Mrs Ward, 66, said she was in a state of disbelief when she first heard what the "kind and charismatic" man she knew was accused of doing.
The married mother-of-four said she is ashamed of Savile's actions, and has no idea what she would say to the victims.
She said: "It was disbelief at first. It's like I've lived a sheltered life and never known any of these things were going on.
"The first week, well I practically spent the first week crying. I've been sent lots of flowers, had phone calls and cards, with people offering support.
"It's nothing to do with me, but in a sense I feel ashamed. I'm sure it's the same with the rest of the family, although I wouldn't like to speak for them.
"Everyone has been talking about all the awful stuff. The good work he committed himself to all his life has been quickly forgotten.
"What can I say, what would you say to the victims if it was your relative?"
To date, officers are following more than 400 lines of inquiry.
Gary Glitter has been arrested in connection with the same investigation.
Lothian and Borders Police are handling claims from women that they were abused by Savile.
Mrs Ward said she and husband Tony watched the allegations with horror when they were first broadcast.
She said: "My husband and I watched the Panorama programme and it was disbelief at first. All these women who have come forward.
"I don't think the family will ever get over it, really."
Mrs Ward was 43 when she discovered Savile was her long-lost uncle. Adopted at 18 months, she waited until her adoptive parents had died before starting to search for her birth parents. It was then she discovered her father was John Henry Savile, known as Johnny, Jimmy's older brother.
Mrs Ward admitted it was still difficult for her to speak ill of Savile, who died aged 84 last year. She said she had only known him as "a kind and charismatic character".
She attended his funeral in Leeds last October and spoke of the support he had given St Columba's Hospice in Edinburgh, where she worked as an auxiliary nurse for 22 years.
She said: "I used to ask him to post his picture and autograph with a brief message for the kids at St Columba's and he would never let us down.
"We posted him a cancer band which he wore to support the hospice. I believe he phoned up to thank us because he couldn't visit in person.
"He came across as an eccentric doing good in public, and to his family he was just a nice man. All that has been destroyed in three or four weeks."
Meanwhile, Edwina Currie, a health minister in Margaret Thatcher's government, has told Channel 4 News that Savile had information "which gave him a hold" over Broadmoor staff after she appointed him to lead a taskforce at the high-security psychiatric hospital.
In a statement, she said: "What he did have, as I know for certain, is information which gave him a hold over staff. That could explain why they said nothing, even with their knowledge or suspicion of his misbehaviour. As a result Ministers were never given the information, when we could have barred him from the place."