The Windrush scandal had heaped pressure on Ms Rudd, who faced renewed criticism after saying she did not know about Home Office removals targets.
Her successor is expected to be announced within hours by Theresa May, who was “very sorry” to see Ms Rudd go.
Shadow home secretary Diane Abbott said Ms Rudd had “done the right thing”.
Ms Abbott added that the “architect of this crisis” – the prime minister – must come before the Commons to explain “whether she knew that Amber Rudd was misleading Parliament and the public last week”.
Ms Rudd told MPs last week the Home Office did not have targets for removing illegal immigrants, but on Sunday the Guardian published a letter in which Ms Rudd set out her “ambitious but deliverable” aim to deport 10% more illegal immigrants over the “next few years” to Theresa May.
Ms Rudd, who had been due to make a Commons statement on Monday afternoon, telephoned the prime minister on Sunday evening to tell her of the decision amid intensifying opposition demands for her to quit.
In her resignation letter, Ms Rudd said she took “full responsibility” for the fact she was not aware of “information provided to (her) office which makes mention of targets”.
In response, Mrs May said she believed Ms Rudd had given her evidence to the Commons “in good faith” but that she understood her decision to resign and take “responsibility for inadvertently misleading the home affairs select committee”.
She should “take great pride” in what she achieved at the Home Office, Mrs May added.