Police leaders were found to have spread false narratives about the disaster, blaming Liverpool fans, and withheld evidence of their own failings.
The legislation, formally known as Public Office (Accountability) Bill, had been delayed by wrangling over how the duty of candour applies to the intelligence services.
Ministers broke the deadlock with a series of amendments approved by MPs on Tuesday, which paved the way for it to clear the Commons. It will now be sent to the House of Lords for further scrutiny before it can become law.
The Conservatives accused Sir Keir of rushing the legislation through its final stages in the Commons following a series of late changes.
Speaking in the Commons, Sir Keir said the Hillsborough families “fought so that no family like them should ever have to endure and suffer what they went through”.
He told MPs: “What they have done will protect individuals and families that they’ll never know, they’ll never meet but who’ll never have to go through what they went through.”
Sir Keir also said: “This is not just a bill for the 97, although it is that, it is a bill for every single working person in this country because, let’s face it, there is a class element to this.
“Time and again the cry for justice was ignored by the British state because of who the victims are – because they’re working class, because they’re black, because they’re women and girls.”
He raised the work of campaigners over failings connected to other scandals, including the Grenfell Tower fire, the Manchester Arena bombings, the infected blood scandal and grooming gangs.
Speaking shortly before the bill was approved at third reading, Sir Keir said: “We stand on the precipice of change. This is a moment of real significance for our country.”
Sir Keir later told a reception attended by campaigners who had lost loved ones that their work had had a “profound impact on me”, adding: “I can tell you that whatever happens next in my life, I’m never giving up on this.”
Burnham, who is on course to replace Sir Keir as prime minister on 20 July, said the “momentous” legislation will “change the way this country thinks and works about justice”.
He said: “It truly is a rewiring of the state and a passing of power from the authorities to the hands of ordinary people.
“And it is happening because of the prime minister’s commitment to a country based on justice and fairness, and we thank him for that.”
Burnham said Sir Keir has “honoured his commitment to the Hillsborough families”, adding the bill will ensure “justice for ordinary people going forward, and that is some legacy indeed”.
