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.
The former Home Secretary was due to have faced the Commons on Monday with opposition MPs accusing her of having misled Parliament after she told a Commons committee last week that the Government did not have targets for removals.
Her position appeared to weaken after the former immigration minister Brandon Lewis disclosed he had weekly discussions with her about how they could get the numbers up when he was in the Home Office.
However, appearing on BBC One's The Andrew Marr Show, Mr Lewis, the current Conservative Party chairman, insisted they had not talked about specific targets.
He was accused by shadow home secretary Diane Abbott of "hiding behind semantics", saying his disclosure made it clear she knew what was going on in the department.
"Beneath the spin, he let the truth slip and sealed her fate. Amber Rudd knew of the targets she pretended didn't exist. It's time for Rudd to go," she said.
Earlier in the day, Downing Street has said she continues to enjoy the "full confidence" of Theresa May while she has received public support from both senior ministers and Tory backbenchers.
Ms Rudd's 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 Commons 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.