Ms Rudd telephoned Theresa May to tell her of the decision amid intensifying opposition demands for her to quit.
A No 10 spokesman said: “The Prime Minister has tonight accepted the resignation of the Home Secretary.”
Ms Rudd was thought to be preparing to tough it out, insisting she genuinely did not know about the targets when she gave evidence to the Commons Home Affairs Committee last week.
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.
Her decision to stand down will come as a major blow to Mrs May who publicly declared her “full confidence” in her as recently as Friday.
Even before she was accused of misleading MPs, she had been under fire for her handling of the Windrush scandal which saw Commonwealth citizens who came to Britain in the post-war decades being wrongly threatened with deportation.
Labour said she was effectively acting as a ‘human shield’ for Mrs May whose policy when she home secretary of creating a ‘hostile environment’ for illegal immigrants was blamed for causing the problems they now faced.
Liberal Democrat home affairs spokesman Sir Ed Davey said: “It’s clear that Amber Rudd has ended up, at least partly, being the fall guy to protect the Prime Minister. Theresa May must face questions now given these dreadful failures largely took place under her watch as Home Secretary.”
Ms 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 Wednesday 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.
Ms Rudd’s immediate difficulties began on Wednesday when she told the Commons Home Affairs Committee that the Home Office did not have targets for removals.
The following day, however, she returned to the House to admit that Immigration Enforcement managers did use “local targets” but she said they were “not published targets against which performance was assessed”.
The pressure then ratcheted up on Friday with the leak of a Home Office memo, which referred to a target of 2,800 enforced returns for 2017-18, and the progress that had been made towards a “10% increased performance on enforced returns, which we promised the Home Secretary earlier this year”.
In a series of late night tweets, Ms Rudd said she had not seen the memo - even though it was copied to her office - but admitted that she should have been aware of the targets.
She said she would be making a Commons statement on Monday to answer MPs “legitimate questions” about what had been taking place.
Her position was further weakened over the weekend however when former immigration minister Brandon Lewis - now the Conservative Party chairman - disclosed they had held weekly meetings to discuss
her plans to increase the numbers of removals, although he insisted they did not discuss detailed targets or numbers.