Cabinet minister Andrea Leadsom has said it is "vital we get justice" for Harry Dunn as his family said they were "incredibly reassured" after meeting the home secretary.
The family will also meet the prime minister in Downing Street in the new year to discuss the case.
Nineteen-year-old Harry died in August when his motorbike was hit by a car driven by Anne Sacoolas, the wife of a US diplomat.
His family met Home Secretary Priti Patel as her office considers extraditing Sacoolas, who has now been charged with causing the teenager's death by dangerous driving.
The Crown Prosecution Service said extradition proceedings had started, noting the "Home Office is responsible for considering our request and deciding whether to formally issue this through US diplomatic channels".
Speaking after the meeting, Dunn family spokesman Radd Seiger said: "We feel incredibly reassured that this whole saga will be dealt with under the rule of law."
Ms Leadsom, who is the Dunn family's constituency MP as well as being business secretary, said: "As the home secretary has made clear we have a very clear process in the UK, there's a clear extradition treaty and it is absolutely vital that we get justice for Harry."
Ms Patel said she was "very grateful" to the family for meeting her during such a difficult time.
She added: "It was a nice opportunity to hear from them, obviously about what they have been experiencing, what they have been going through, and to reassure them at what has been a very difficult and traumatic time for them."
Sacoolas, the wife of an American intelligence officer, left the UK after the crash outside an RAF base in Northamptonshire.
The 42-year-old sparked claimed diplomatic immunity through her husband's job.
It was only after she had left the UK on a military flight that the Foreign Office wrote to the 19-year-old's family to say she did not have immunity.
Following the charging decision, Sacoolas' lawyer Amy Jeffress said her client had "co-operated fully with the investigation" and would not return voluntarily to the UK.
The State Department has also said the extradition of a former diplomat's wife to the UK would be an "egregious abuse".
Mr Seiger said the family was angry Sacoolas was back in the US and felt she should never have been allowed to leave.
He added: "They don't understand why this lady is back in America, they feel she should never have been allowed to go and should jolly well come back.
"This lady is absolutely entitled to a fair trial, she is guilty of nothing at the moment. If she comes back, when she comes back she will get a fair hearing."
Commenting on the case during a visit to Estonia to see British troops, Boris Johnson said: "The law should take its course and we'll be following the case with keen interest and continuing to make representations on behalf of Harry Dunn's family in the US at every level."