Harry, 19, was killed when his motorbike crashed into a car outside a US military base in Northamptonshire on August 27 last year.
Ms Sacoolas, 43, the wife of a US intelligence official and a reported CIA operative, claimed diplomatic immunity following the crash and was able to return to her home country, sparking an international controversy.
Foreign Secretary Dominic Raab revealed on Wednesday that changes had been made that would “see justice done” if another family was to end up in a similar position to that of the Dunn family.
Despite the “anomaly” being amended in the immunity agreements surrounding RAF Croughton, the base near where Harry died, his alleged killer still remains in the US.
Mr Raab said in a written statement: “First and foremost, the US waiver of immunity from criminal jurisdiction is now expressly extended to the family members of US staff at the Croughton Annex, thus ending the anomaly in the previous arrangements and permitting the criminal prosecution of the family members of those staff, should these tragic circumstances ever arise again.
“We have the deepest sympathy for Harry Dunn’s family. No family should have to experience what they have gone through and I recognise that these changes will not bring Harry back.
“However, I hope that the knowledge that the Croughton arrangements have been revised and that a family in their position would now see justice done brings some small measure of comfort.”
Harry’s mother Charlotte Charles told the PA news agency, Wednesday’s announcement was a “huge step forward” – adding that one of the family’s aims was for this to “never happen to another family again”.
She said their campaign would continue for Sacoolas’s return to the UK.
Timm Dunn, Harry’s father said the changes were “great news” in a post on Twitter.
“Should stop what happened to us happening to anybody else,” wrote Mr Dunn in the post.
The US State Department said the amendment of the diplomatic immunity arrangements at RAF Croughton “is a reflection of our especially close relationship” with the UK.
A spokesman said: “Under the arrangement, the United States has extended pre-emptive, limited waivers of certain diplomatic immunities pertaining to the staff of the Embassy office in Croughton and their family members for acts performed outside the course of official duties that occur on or after July 20, 2020.
“This arrangement is a reflection of our especially close relationship with the government of the United Kingdom.”
Northamptonshire Police said while the changes were not retrospective the force “welcomes these changes”.
“Northamptonshire Police has today been advised by the Foreign and Commonwealth Office that they have reached an agreement with the US government to restrict the immunity from criminal jurisdiction of embassy staff, officers and family members at RAF Croughton,” said the police force in a statement.
“Northamptonshire Police remains committed to working with colleagues in the Crown Prosecution Service to ensure Anne Sacoolas is returned from the US to allow criminal proceedings to go ahead here in the UK.”
An extradition request submitted by the Home Office for Ms Sacoolas was rejected by Mike Pompeo in January – a decision the State Department has said is “final”.
Ms Charles urged US and British leaders to discuss the case during Mr Pompeo’s visit to London on Monday.
The Dunn family want a “very clear timetabled structured plan to bring Anne Sacoolas back,” their spokesperson Radd Seiger told Sky News as he called on Mr Raab to “stand up to the US”.
He told the programme that “It is no longer any good … that they raise it continually. That means nothing to the family.”