The pair have both now walked away from Downing Street as the PM turns her attention to trying to strike a deal with the Democratic Unionist Party.
Mrs May fell eight seats short of a House of Commons majority and she will now be reliant on the DUP's 10 MPs to add to the 318 Tories in order to get the 326 seats needed to pass legislation.
Announcing his decision to quit in a piece on the Conservative Home website, Mr Timothy said he took "full responsibility" for his role in the campaign and expressed regret at the way in which social care plans, dubbed a "dementia tax" by critics, had been rolled out without including a cap on costs.
He said: "I take responsibility for my part in this election campaign, which was the oversight of our policy programme.
"In particular, I regret the decision not to include in the manifesto a ceiling as well as a floor in our proposal to help meet the increasing cost of social care.
"But I would like to make clear that the bizarre media reports about my own role in the policy’s inclusion are wrong: it had been the subject of many months of work within Whitehall, and it was not my personal pet project.
"I chose not to rebut these reports as they were published, as to have done so would have been a distraction for the campaign. But I take responsibility for the content of the whole manifesto, which I continue to believe is an honest and strong programme for government."
Nick Timothy and Fiona Hill pictured leaving Conservative Party HQ on Friday Credit: Frank Augstein/AP
Mr Timothy also rejected the idea that the Tories had failed at the election as he pointed out that the Tories had won almost 14 million votes.
However he did admit that Mrs May's failure to secure a majority was "disappointing" but blamed that failure on an "unexpected surge in support for Labour".
Ms Hill also released a statement on Conservative Home and she said: "It's been a pleasure to serve in government, and a pleasure to work with such an excellent Prime Minister.
"I have no doubt at all that Theresa May will continue to serve and work hard as Prime Minister - and do it brilliantly."
The BBC reported that senior Tories had said the pair would have to be sacked this weekend or that Mrs May would face a leadership challenge on Monday.
The resignations came as Downing Street confirmed that Gavin Williamson, the chief whip, had been sent to Northern Ireland to try and hammer out a deal with the Democratic Unionist Party to keep Mrs May in power.
A Downing Street spokesman said: "The Chief Whip is in Belfast holding talks with the DUP on how best they can provide support to the Government. We will not be providing a running commentary."
Mr Williamson's visit comes amid reports that he will try to strike a formal coalition deal with the DUP to stop them holding the Tories to ransom on an issue by issue basis instead of a more informal arrangement.
ITV political editor Robert Peston said ministers had told him that Mrs May will seek to persuade the DUP's 10 MPs to enter into a formal coalition instead of simply backing the Tories on individual votes.
Meanwhile pressure continues to grow on the Prime Minister's position with almost two thirds of Conservative Party members believing that she should resign and trigger a leadership contest.
A snap survey conducted by the Conservative Home website revealed that 60 per cent of party members believe Mrs May should quit while just 37 per cent of the 1,500 members who took part said she should stay in post.
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