It appears that Rudd took the decision to quit herself, despite Downing Street previously trying to prop her up. She telephoned the PM to inform her of the move.
In her response to Rudd, May said she was "very sorry" to receive the resignation. The PM said she still believed Rudd had answered questions from MPs in "good faith".
Tories expressed sadness at the departure - while opposition parties wasted no time in turning their fire on May.
Labour deputy leader Tom Watson said: "I see Amber Rudd is carrying the can for the person originally responsible for this scandal - Theresa May."
The timing of the resignation took Westminster by surprise. There was intense speculation that Rudd might go last weekend after the emergence of a leaked Home Office memo that had been copied to the minister.
It spelled out that there were both national and regional targets for deportations of illegal immigrants - seemingly contradicting the evidence Rudd gave to the Home Affairs select committee a day earlier.
But after hours of ominous silence from the Home Office, Rudd broke cover to insist she would stay on.
She vowed she genuinely did not know about the targets when she gave evidence to MPs - and said she would make her case to the Commons in a statement tonight.
However having seen mounting evidence in the paperwork about the extent of the knowledge within the Home Office about the targets she decided that she should take responsibility and go.
It comes after another private letter which included "ambitious and deliverable" migrant deportation targets emerged, after Rudd claimed she knew nothing about them.
Rudd appeared to have signed the correspondence, which said her department aimed to "increase the number of enforced removals by more than 10 per cent", in January last year.
The Home Secretary had already claimed that she had never seen a previous memo referencing immigration targets - and the letter appears to have been the final straw.
The blunder led to Labour calls for her resignation and growing disquiet among her colleagues.
Allies of Rudd said the former Home Office Secretary felt isolated by No 10 and let down by her officials.
One said: "Amber has been caught in a s*** sandwich. There has been no support from Downing Street, either politically or in terms of communications."
The six-page page memo prepared by Hugh Ind, the director general of Immigration Enforcement in the Home Office used policy ideas outlined by Rudd in her private letter.
Included in the document, leaked to the Guardian, were targets such as "achieving 12,800 enforced returns in 2017/18."
The note was also addressed to Marc Owen, senior director of national and international operations in Immigration Enforcement, Mark Thomson, the director general of the Passport Office and Tony Eastaugh, UK director of operations at Immigration Enforcement.
Rudd's departure will also upset the delicate balance within the Cabinet between Leavers and Remainers ahead of a crucial meeting of the Brexit "war cabinet" on Thursday to discuss Britain's future customs relationship with the EU.
Environment Secretary Michael Gove - one of the Cabinet's leading Brexiteers - was being touted as the front runner replace her at the Home Office.
Shadow Home Secretary Diane Abbott said: "Amber has done the right thing. The Windrush generation could not have had faith in her. She made promises she brushed under the carpet. It is a Home Office scandal."
Abbott told BBC Radio 4's Westminster Hour: "Politics is very hard and the only sympathy goes to the generation of people who have been let down by this Government. We need justice."
Labour MP David Lammy, a leading campaigner on Windrush, tweeted: "Amber Rudd resigned because she didn't know what was going on in her own department and she had clearly lost the confidence of her own civil servants.
"The real issue is the hostile environment policy that caused this crisis in the first place.
"The resignation of the Home Secretary must not detract from the fact that this crisis was a direct result of the hostile environment policy.
"That policy must now be reviewed, and the Home Office must move quickly to compensate and grant citizenship to the Windrush generation."
In a sign that Rudd could join Remainer rebels on the Tory backbenches, former minister Anna Soubry said: "V sorry that @AmberRuddHR has resigned. She is a woman of great courage & immense ability.
"Amber will be missed in many ways. We'll give her a huge welcome on to our back benches. If there is any justice she will soon return to the highest of office. Proud to call her my friend."
Lucy Frazer, Tory MP for South East Cambridgeshire, added: "I'm very disappointed she felt she had to resign. Amber was an excellent Minister... She's put a number of measures in place for the Windrush generation."
RUDD FIFTH TO GO
Five Cabinet members have left their positions in the 10 months since June's snap general election where the Conservatives lost their majority in the House of Commons.
The first to leave was Defence Secretary Michael Fallon, who resigned his post after being caught up in Westminster sleaze allegations.
He admitted his standards had "fallen below the high standards required" after he admitted putting his hand on the knee of radio presenter Julia Hartley-Brewer some years ago when he resigned on November 1.
A week later, Priti Patel quit as international development secretary over secret meetings in Israel, including with Prime Minister Benjamin Netanyahu.
In her resignation letter, she echoed the words of Fallon, saying her actions "fell below the high standards" expected.
Theresa May suffered a third Cabinet loss in seven weeks when Damian Green left as First Secretary of State in December.
While serving as May's de facto deputy, Green made "misleading" statements about allegations that police found pornography on computers in his parliamentary office in 2008 which breached the ministerial code.
Northern Ireland Secretary James Brokenshire left the Cabinet in January after standing down on health grounds, and Rudd became the fifth Cabinet member to leave amid claims she misled Parliament over targets for removing illegal migrants.
