In a letter to the PM he said the government “owe it to people who have sacrificed so much in this pandemic to be honest when we have let them down”.
Boris Johnson said he was “sorry” to receive the resignation.
It comes after pictures emerged of him with Gina Coladangelo, reportedly taken on 6 May.
Mr Hancock had been under increasing pressure to quit, after The Sun published pictures of Mr Hancock and Ms Coladangelo, who are both married with three children, kissing. The newspaper said they had been taken inside the Department of Health.
Fellow Tory MPs, as well as Labour and the Covid-19 Bereaved Families for Justice group, had called for the health secretary to be sacked.
BBC political editor Laura Kuenssberg said No10 had stressed that it had been Mr Hancock’s decision to go and that he had not been pushed out by the prime minister.
She said Ms Coladangelo had also left her role as a non-executive director of the Department of Health.
In a video posted on Twitter, Mr Hancock said: “I have been to see the prime minister to resign as secretary of state for health and social care.”I understand the enormous sacrifices that everybody in this country has made, that you have made, and those of us who make these rules have got to stick by them and that’s why I have got to resign.”
In his resignation letter, Mr Hancock reiterated his apology for “breaking the guidance” and he apologised to his family and loved ones for “putting them through this”.
In response, the prime minister said Mr Hancock “should leave office very proud of what you have achieved – not just in tackling the pandemic, but even before Covid-19 struck us”.
He added: “I am grateful for your support and believe that your contribution to public service is far from over.”
Labour leader Sir Keir Starmer tweeted: “Matt Hancock is right to resign. But Boris Johnson should have sacked him.”
The SNP’s Westminster leader Ian Blackford tweeted: “Massive failure of leadership by Boris Johnson, Hancock should have been sacked.”
Liberal Democrats leader Sir Ed Davey tweeted that Mr Hancock’s legacy was one of “failure” and he said the fact that Mr Johnson “could just carry on regardless brings the prime minister’s judgement into question once again”.
Concerns had also been raised about the process which saw Ms Coladangelo, a friend of Mr Hancock’s from their days at Oxford University, appointed to the role – last September.
A No 10 spokesman has insisted the “correct procedure” had been followed but refused to go into detail.