The report concluded that there was a "pernicious and institutionalised racism" among the Metropolitan Police and implied that the force's Commissioner, Sir Paul Condon, faces the sack unless he accepts its findings.
Mr Justice Rix last night ruled that extracts of the report which had appeared in early editions of the Sunday Telegraph could be re-published and commented upon by other newspapers and broadcasters.
The ruling meant a partial lifting of an injunction obtained less than 24 hours earlier on Mr Straw's instructions, banning any publication of material from the report ahead of its official release on Wednesday.
According to the leaks, the report of Sir William Macpherson of Cluny, who carried out the inquiry, concluded that "institutionalised racism" in the Metropolitan Police had been a key factor in its failure to catch the black teenager's killers.
They also confirm a report in The Independent last week that Sir Paul would be criticised for failing to realise there were serious problems with the investigation and for a "somewhat less positive approach" than other police chiefs who have accepted racism is ingrained in the service.
The Macpherson report will include 70 recommendations affecting policing and race relations in Britain. A coalition of national newspapers, led by the paper's publishers - the Telegraph Group - and The Independent, agreed a variation to the injunction with Treasury solicitors which allows reporting of matters already in the public domain.
Tories seized upon the lifting of the ban on publication of leaked extracts from the inquiry report as a "humiliating defeat" for the Government. Sir Norman Fowler, the shadow home secretary, said the Government should never have sought the High Court injunction in the first place.
"The Government has been forced to climb down. I think it is a humiliating defeat," he said. "The Government should never have taken this to a judge and sought an injunction in the way that they did.
"They acted in an arbitrary and a high-handed manner and they came a cropper and they deserved to have done so."
A Scotland Yard spokesman said last night: "The Commissioner has never said that his future is in Jack Straw's hands; that is not a quote from the Commissioner. "His position is our official position - which is that neither he nor the Metropolitan Police have seen the report and we feel it is inappropriate to comment until it is laid before Parliament."
Chief Superintendent Des Parkinson, national secretary of the Police Superintendents' Association for England and Wales, welcomed the lifting of the injunction but said he was "sad" that the report had been leaked.
Referring to the criticisms of Metropolitan Police Commissioner Sir Paul Condon in the report, Mr Parkinson said: "We are not happy about the criticisms of Sir Paul. We had a meeting and there was unanimous support of Sir Paul, saying he should stay when this report came out.
"We don't believe that there is institutionalised racism in the Metropolitan Police force or in the police service in general."
However, the Home Office played down the ruling and insisted Mr Straw remained unchanged by it. "It is his firm belief that the partial leaking of the report was unfair to the Lawrences, the police and to Parliament," the Home Office said in a statement.
"The principle is upheld by the maintenance of the injunction against publication of any further material from the report.
"Today's variation of the injunction represents no more than a recognition of the practicalities of the situation, namely that some of the report's contents are now in the public domain. But clearly the principle that the report should not have been disclosed remains intact."Reuse content