He also confirmed the BBC would conduct an inquiry following a police investigation.
He spoke out a day after Prime Minister David Cameron called for allegations of sexual abuse against the late celebrity to be fully investigated.
Mr Cameron said that the claims, made by a number of women over recent weeks alleging Savile abused them as girls, were "truly shocking".
And he said that the allegations should be looked into by the BBC – which employed Savile at the time – and, if necessary, by the police.
Mr Entwistle said there needed to be a "comprehensive examination" of what went on, following the police investigation.
"These are awful allegations that have been made and they are criminal allegations, " said Mr Entwistle.
He continued: "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."
He added: "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."
Any BBC probe, he added, would examine the "broad question of what was going and whether anybody around Jimmy Savile knew what was going on".
Mr Entwistle added that he was told that Newsnight was looking at a possible investigation into Savile at around early December last year.