A leaked memo to the Guardian claimed Amber Rudd was included in a message past year detailing targets for migrant removal.
Her tweets came hours after the leaked memo was revealed and political journalists waited on tenterhooks to see if she would resign.
She has been accused of misleading Parliament after denying the Home Office used targets for removals.
Ms Abbott said: Ms Rudd's failure to read "crucial documents" meant she was not aware of targets that have "led to people's lives being ruined".
Rudd said she would make a statement to the House of Commons on Monday to answer "legitimate questions that have arisen on targets and illegal migration".
She later admitted "local" targets had been set before telling the Commons on Thursday she had not been aware of them.
UK Home Secretary Amber Rudd on Friday apologized for not being aware of her department's migrant removal targets but rejected the calls to resign over policy failings. I accept I should have been and I'm sorry that I wasn't.
However she acknowledged that the disclosure had raised "legitimate questions" and said she would be making a statement to MPs on Monday.
In the third tweet, the Home Secretary said: I didn't see the leaked document, although it was copied to my office as many documents are.
Just over half - 53% - think the Government should set targets for the annual removal of illegal immigrants, while 39% think there should not.
The "hostile environment" policies implemented by the Home Office, created to discourage illegal immigrants from staying in the country, have been blamed for the scandal.
Earlier, shadow home secretary Diane Abbott said: "Amber Rudd either failed to read this memo, and has no clear understanding of the policies in her own department, or she has misled Parliament and the British people".
A spokesman for Downing Street said: "The PM has full..."
The public are also split on Home Secretary Amber Rudd's future - 41% think she should resign and 41% think she should stay in her job, with 18% answering "don't know".
She told MPs after it was revealed the Home Office did in fact have targets: "Some offices are working with them".
But bookmakers have already installed Michael Gove, the Environment Secretary, as favourite to succeed her, with Housing Secretary Sajid Javid and James Brokenshire - a former Home Office Minister who quit as Northern Ireland Secretary to have lung surgery, but is now recovering - also being tipped.
Nor will she want such a prominent Remainer on her back benches - someone who could become a major focal point for Tory rebels - with key Commons votes on Brexit coming up.