Ms Rudd told a parliamentary committee last week that the Government had no specific targets for the removal of illegal immigrants.
But critics claimed she misled Parliament when a leaked Home Office memo referred to a target of 2,800 enforced returns for 2017-18.
And tonight it was announced that Prime Minister Theresa May had accepted Ms Rudd’s resignation after almost two years in the role.
A Downing Street spokesman said: “The Prime Minister has tonight accepted the resignation of the Home Secretary.”
The Remain campaigner is the fourth Cabinet minister to quit in just six months, following Damian Green, Priti Patel and Sir Michael Fallon.
Responding to Ms Rudd’s resignation tonight, Tory backbencher Anna Soubry tweeted: “Very sorry that Amber Rudd has resigned.
“She is a woman of great courage and 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.”
Andrea Leadsom, leader of the House of Commons, added: “Genuinely sorry to see my friend Amber Rudd resign. She is a very honest and principled person.”
Damian Green added: “Very sad news about Amber Rudd. She is a decent, clever and conscientious woman who has so much to offer our politics.
But Labour’s deputy leader Tom Watson said: “I see Amber Rudd is carrying the can for the person originally responsible for this scandal – Theresa May.”
And Liberal Democrat MP Layla Moran said: “The Government have no plan and no clue.
“The resignation of the Home Secretary highlights the total chaos they are in. To be frank they are rudderless.”