Mr Entwistle announced that two "soul-searching" investigations would be carried out into allegations the disgraced former TV presenter and DJ abused children at its studios, and to establish why a Newsnight investigation that may have exposed his "illegal activities" sooner was dropped.
Scotland Yard, which is leading the criminal investigation, said the number of lines of inquiry had almost tripled since Tuesday. It has described Savile as a sexual predator on a "national scale".
The BBC investigations, which will be independent, will also examine the broadcaster's policies on child protection, bullying and whistleblowing. BBC Scotland director Ken MacQuarrie is to interview Newsnight staff and record their opinions as to why the planned report on Savile was dropped.
Mr Entwistle said: "I have made clear my revulsion at the thought these criminal assaults were carried out by someone employed by the BBC and that some may have happened on BBC premises as well as, we now discover, in hospitals and other institutions across the UK. I have one thing to repeat – that is a profound and heartfelt apology on behalf of the BBC to every victim.
"It is the victims, these women who were subject to criminal actions, who must be central in our thoughts."
He added: "These will be forensic, but also soul-searching investigations. Our audience's trust in us is paramount. We will do everything in our power to maintain that trust and we will do that by holding ourselves to account fully and openly.
"Jimmy Savile's victims have faced years of pain. We owe it to them and to our audiences to understand how this could have happened and to make sure that everything we do ensures nothing like this can ever happen again."
The scandal has seen the BBC come under fire amid allegations the corporation was aware of the former Jim'll Fix It and Top of the Pops presenter's behaviour and failed to take action.
David Nicolson, who worked as a director on the children's programme, claimed to have caught Savile having sex with a girl in his dressing room, but said he was laughed at when he voiced his concerns.
He said: "It was a bog-standard changing room in the basement. The girl could have been 16, maybe 15. But she was just one of many – he always had one in the room."
When he reported the incident, he said he was told: "That's Jimmy." He said: "I was revolted by his behaviour. They just shrugged it off, saying 'Yeah, yeah. That's the way it goes'." Mr Nicolson added: "Everyone knew what was going on. That includes senior BBC people – chiefs at the highest levels."
The National Society for the Prevention of Cruelty to Children (NSPCC) said calls to its hotline from alleged victims of sexual abuse had doubled since the revelations about the former DJ became public.
Since ITV broadcast a documentary about the claims, it has received 100 calls from people wishing to discuss sexual abuse. That figure does not include another 40 about Savile.
Allegations against Savile include that he targeted patients at Stoke Mandeville in Buckinghamshire and Leeds General Infirmary.
Leeds Safeguarding Children Board said it had not received any complaints about Savile, but encouraged anyone with concerns to report them.
Matt Forde, head of services for the NSPCC in Scotland, said: "While Jimmy Savile can obviously never be tried for these alleged crimes, we can support the victims and identify lessons learned to prevent this from happening again. To do this we must have a full picture of what happened and reach all the victims involved.
"It's vital that both victims and others with information about sexual abuse come forward to offer information or seek help, no matter when the offence was committed or who the offender was."
Deputy Prime Minister Nick Clegg said he thought the case exposed the dark side of Britain's celebrity culture. "I have been thinking about this a lot because I just cannot understand how this remained hidden for so long," he told BBC Radio Five Live.
"I just keep asking myself 'Why did this remain buried for so long?' The only explanation I can come up with is what we are seeing is the dark side of the culture of celebrity, and actually in this case it wasn't a culture of celebrity, it was the cult of celebrity."
Actress Julie Fernandez said she had been a victim of Savile's when she appeared on Jim'll Fix It aged 14.
The disabled performer, now 38, who appeared in the BBC's Eldorado and The Office, said his hands had lingered "in places they shouldn't".