Theresa May's promise of a review of the Windrush generation scandal helped see off Labour attempts to force her Government to publish eight years of internal documents.
The Prime Minister pledged a "package of measures to bring transparency on the issue", which included the review having "full access" to Home Office information.
Home Secretary Sajid Javid later told MPs the plans included him writing each month to Yvette Cooper, chairwoman of the Home Affairs Select Committee, to provide updates on "progress" along with the latest position on detention, removals and deportations.
He added the "lessons learned" review would have "independent oversight and challenge", with its aim to understand how members of the Windrush generation "came to be entangled in measures designed for illegal immigrants", why it was not spotted sooner and if the "right corrective measures" are in place.
The Government's proposals came as it faced being forced to hand over correspondence - including text messages and emails - between ministers, officials and special advisers between May 2010 and 2018.
Labour used a parliamentary procedure - which involves asking the Queen to direct her ministers to provide the requested documents - to try and ensure the documents would have been given to the Home Affairs Select Committee.
But the Opposition's motion was comfortably defeated by 316 votes to 221, majority 95.
Shadow home secretary Diane Abbott accused Mrs May of ordering her MPs to "vote to cover up the truth of her involvement".
She said in a statement: "This is nothing short of a betrayal of the Windrush generation and others who have been affected by the Government's heartless 'hostile environment', and flies in the face of the new Home Secretary's promise to make things right.
"Tory MPs will now have to explain to the people whose lives have been turned upside down why they think they don't deserve the facts to be known and don't deserve proper justice."
Ms Abbott earlier told the Commons that many people felt "all roads lead back to the Prime Minister" on the Windrush scandal, with Mrs May having previously held the post of home secretary.
Ms Abbott added that until the Prime Minister announced an end to the "hostile environment", the policy was still in place.
In stormy scenes in the Commons, Mr Javid accused Labour of a "fishing expedition" and claimed complying with the motion would have resulted in more than 100 officials being reassigned - which would have taken away help and support from the Windrush cases.
He said: "I know I'm not alone when I say there are men and women from the Windrush generation who have been seriously let down by the immigration system - men and women who have had their lives totally and utterly disrupted and put on hold in many cases."
Mr Javid reiterated he would do "whatever it takes" to put it right.