Mr Entwistle, who resigned on Saturday over the botched Newsnight programme on child abuse, has been awarded a full year's pay, despite normally being entitled to only half that figure.
Lord Patten, the BBC Trust chairman, defended the sum, saying it was justified and necessary to allow a clean break and avoid lengthy delays.
However, the Prime Minister's spokesman made clear the pay-off was hard to justify and, in an emergency Commons statement, Maria Miller, the Culture Secretary, described the severance payment as a reward for failure but added that ultimately it was a matter for the BBC Trust to decide the figure.
The payment was described as outrageous by some MPs yet in a letter to John Whittingdale, chairman of the Commons Culture, Media and Sport Committee, Lord Patten insisted it was agreed to avoid "long drawn-out discussions and continuing uncertainty".
He wrote: "In circumstances where we needed to conclude matters quickly and required George's ongoing co-operation in a number of very difficult and sensitive matters, including the inquiries into issues associated with [DJ Jimmy] Savile, I concluded a consensual resignation on these terms was clearly the better route.
"I consulted my colleagues on the trust's remuneration committee and took legal advice. Our conclusion was that a settlement on these terms was justified and necessary."
While some MPs called on Lord Patten to resign, others argued he should stay to oversee the rebuilding of public trust in the Corporation.
The National Audit Office said it would be talking to the BBC Trust about Mr Entwistle's pay-off to see if it represented good value for licence fee payers.
Former Tory Party deputy chairman Lord McAlpine was linked to the Newsnight story on the internet, even though the allegations about a senior Tory being involved in sex abuse were untrue. Tim Davie, the acting director-general, who promised to "get a grip" on BBC management, said he wanted to apologise personally to the Tory peer.
He addressed staff by email, telling them he was "determined to give the BBC the clarity and leadership it deserves", and said management would pull together as one team.
Elsewhere, Mr Whittingdale said the "monumental failures" at the BBC had to prompt a complete overhaul of its management structure. He claimed the crisis at the Corporation was so serious because it had undermined public trust in traditional news outlets.
Meanwhile, last night, Mr Cameron got a laugh at his annual Guildhall speech in the City of London with a joke about the furore engulfing the Corporation. "Tonight, I want to make a different kind of speech," he said, sticking to his script, before adding: "Don't worry, it's not about the BBC."
Today, MPs will debate the issue of child sexual exploitation.