George Entwistle, the new director-general of the corporation, made the pledge as he offered the first apology to the victims of the late DJ and presenter. Just days earlier he had ruled out an internal inquiry on the grounds that it was up to police to look into criminal allegations.
He also said he was sorry that BBC 2's flagship current affairs programme, Newsnight, had not broadcast the documentary it planned on Savile.
The public service broadcaster has come under pressure to hold an investigation after close to 100 people came forward to say they had been molested by Savile when young, some in BBC dressing rooms.
Some former staff have claimed that the presenter's behaviour was well-known but was tolerated either because he was such a big star or because he raised so much money for charity.
One Conservative MP has asked Lord Justice Leveson to look at how the BBC dealt with the allegations in his public inquiry into media standards, while another called for an independent investigation in a request rejected by the chairman of the BBC Trust.
Speaking on Radio 4's Today programme on Monday morning, Mr Entwistle said: "These are awful allegations that have been made, and they are criminal allegations.
"And the first thing I want to say is that the women involved here have gone through something awful, something I deeply regret they should have to go through, and I would like to apologise on behalf of the organisation to each and every one of them for what they've had to endure here."
Scotland Yard is now sifting through reports of crimes made by women across the country, and Mr Entwistle said: "When the police have finished everything they have to do, and when they give me an assurance that there is no danger of us in any way compromising or contaminating an investigation, I will take it further and ensure that any outstanding questions are answered properly."
The director-general said Savile was regarded as a "bit peculiar" but if anyone at the BBC had evidence of particular allegations there would have been an "enormous obligation" on them to take action.
A former head of the BBC's light entertainment group, Jim Moir, has also insisted that he never heard abuse claims about Savile.
He told The Daily Telegraph: "During my time as head of the department, neither from an external source nor internally, by nod, wink, innuendo or inference, either directly or anonymously, did any allegation of this nature ever come to me."
Mike Read, a Radio 1 DJ who attended Savile's funeral, said: "No one knew about his private life. We only knew him as a senior broadcaster. Most of us at Radio 1 probably only saw him one or twice a year as he was a separate entity."
Johnny Beerling, the former controller of Radio 1 who sacked Savile, said: "Jimmy Savile was a loner. He rarely socialised with any of the DJ's or staff at Radio 1 and as the title of his programme, 'Savile's Travels' suggests it was recorded as he travelled around the country mostly away from BBC premises.
"I was not aware of any sexual improprieties which have now been uncovered."