Aman Khalili and his family had been fearing for their lives and were in hiding from the Taliban after the insurgents seized control of the capital of Kabul in August.
Mr Khalili said he and his family were evacuated from Afghanistan by the Human First Coalition and the US Department of State and have since left Pakistan.
The father told ITV News "now I feel safe" and said he was "thankful" to President Biden and to others "who participated in rescuing me and my family from the dire situation in Afghanistan".
Mr Khalili had previously issued a direct plea to Mr Biden in an interview with ITV News to return the favour and rescue him and his loved ones from "hell", as he struggled to secure a US visa.
He today told of the heartbreak and "frustration" he and his children faced when they were unable to board a flight after borders closed and US forces withdrew in late August.
The family were prepared to evacuate after receiving a call from a colleague who said he could get them on a flight from Hamid Karzi Airport. "We all were ready and packed," recounted Mr Khalili, but when he was asked how many were in the group, they were turned away.
"I answered him we are 13 people. Then he said sorry I cannot."
Mr Khalili pleaded with him but was told the "doors are sealed" and the family were left behind sobbing with their packed bags.
He told his story to US veteran Brian Genthe, who told him "be patient and don't give up".
Mr Genthe went on to help rescue the family and told ITV News of his "relief" that they were safely out of the country.
A senior State Department official would not reveal where the family was heading but told CNN on Monday "we have worked to arrange" their onward travel.
The official confirmed that Mr Khalili's Special Immigrant Visa (SIV) application to the US was rejected several years ago - but they are working "to expedite that reprocessing".
"We engaged at senior levels to help facilitate all of this. We're gratified that he'll be on his way to safety," said the official.
The Human First Coalition, a team of volunteers helping American citizens and allies to safely leave Afghanistan, said they were "grateful" to US and Pakistani leaders for evacuating the family.
Mr Khalili had helped rescue Mr Biden and two other US senators in 2008 after their helicopter was forced to make an emergency landing in the Afghan mountains during a life-threatening snowstorm.
A team of American troops along with Mr Khalili travelled to rescue then-senator Mr Biden, Chuck Hagel and John Kerry - all of them risking their lives to save the future president and his colleagues.
For more than 24 hours, in freezing conditions, they stayed with the senators in Taliban territory until the helicopters could take off to safety at the Bagram military base.
Mr Biden had spoken many times of the traumatic event during campaign trails
Mr Khalili, speaking then under the condition of anonymity to ITV News, had hoped the US president would help him in his hour of need.
He said in September: "My message to the president of the United States and his administration is that they've gone and they've left me behind.
"I request please do not forget me and my family. They have to rescue me and my kids from this hell."
Retired Army Captain Dennis Chamberlain, who served with Mr Khalili, said he was horrified that he was left behind and had vowed to "honour" his service and get him out.
Several US veterans told ITV News of their guilt at abandoning Afghan soldiers who "were fighting and dying right beside us" after the evacuation deadline.