But the memorandum of understanding which allowed the transfer is not legally binding and there is friction between Kabul and Washington over its interpretation.
Six hundred Afghan inmates held since the signing of the documents on March 9 are yet to be handed over to the Afghan authorities.
Karzai's spokesman, Aimal Faizi, insisted that "these people were being held illegally and contrary to the agreement".
Non-Afghan detainees will remain under the control of US authorities and their fate is yet to be decided.
Bagram in many ways encapsulates the tribulations and triumphs of the West and other current interlopers in Afghanistan.
It was built by the Russians and abandoned by Moscow as it pulled its troops back to a collapsing Soviet Union.
Small numbers of Pakistani troops, out of uniform, were briefly stationed there while destroying the ancient irrigation system of the Shomali Plain to halt an advance by the Northern Alliance against their client Taleban fighters.
British troops, first sent in during the 2001 war, were at Bagram, which was soon under sole American control and the centre of Afghan operations, with a degree of success.
At the time, the Taleban were on the run and the warlords, including those from the Northern Alliance, the West's partners in the war, were resigned to losing their private armies and curbing their opium trade.
Teams of intelligence agents from the CIA, MI6 and other Western agencies, veterans of the Mujahideen war against the Russians, were tracking down Osama bin Laden and al-Qaeda, and Mullah Omar's Taleban leadership.
In 2003, the Bush Administration's "War on Terror", with Britain's Tony Blair following, moved to Iraq and so did large numbers of troops.
Bagram emptied, the intelligence agents, with their deep experience of South Asia but not much of the Middle East, were sent off to hunt Saddam Hussein and the Baath leadership.
The Taleban, fed and watered in Pakistan, moved back across the border into the security vacuum.
The recent history of Bagram is enmeshed in allegations of torture,with the detention facilities likened to Abu Ghraib in Iraq and to Guantanamo Bay.
In 2005, a leaked US Army report revealed that military coroners had ruled that the deaths of two prisoners were homicides. Allegations of other abuse followed.
The handover comes as international forces are scaling down their presence.
However, the central element of the withdrawal, Afghan forces taking over security, has itself become one of the most serious sources of concern for the West, with a seemingly unceasing death toll from rogue Afghan troops turning their guns on their allies.
A number of precautionary measures have been introduced, and the Americans have temporarily suspended their training of the Afghan Local Police, a tribal militia which had proved to be effective against the insurgents.
The Americans will retain use of Bagram, along with a number of other bases across the country, as part of a security agreement for another decade after 2014.
The US is expected to provide air support to Afghan troops and, with Britain and other Western countries, station special forces.
The commitment, which was partly dependent on the prison handover, is viewed by Afghans - who fear an attempt by the insurgents to take over after the withdrawal of Western troops - as an insurance policy.
A report commissioned by the authoritative think-tank the Royal United Services Institute said that the Taleban leadership might be prepared to negotiate a ceasefire and even accept a continuing US presence until 2024 at five bases in the country.
The claims were made by current and former Taleban officials during meetings in a Gulf state. It appears, however, that the officials also said the Americans would be vastly limited in what they would be allowed to do - this would certainly not include carrying out more drone strikes inside Pakistan, which have resulted in the deaths of a vast number of insurgent leaders, including from al-Qaeda.
Some "moderate" Taleban leaders are said to have insisted that, once a peace deal is signed by Omar, even the hardline Haqqani network - which the head of the US military described as "a veritable arm" of the Pakistani intelligence service, the ISI - would come on board.
Its head, Jalaluddin Haqqani, can "become a kind of [British Deputy Prime Minister ] Nick Clegg to Mullah Omar's [British Prime Minister ] David Cameron", according to one Taleban official.
"However some in the insurgency will never give up, they are waiting for the coming of [the prophesied saviour of Islam] the Mahdi," said a long-time Afghan watcher and academic.
"Nick Clegg versus the Mahdi, what a contest for the future of Afghanistan," said Jawad Mohammed Nasruddin, a political analyst in Kabul.
"The reality is that neither the Taleban nor the West can believe they have won, but neither side believe they have lost, either.
"We have a long way to go yet before this is finished. What will happen at Bagram, will treatment of prisoners get better there just because Karzai's people have taken over?
"We shall have to see."
Bagram timeline
December 2001
Nato takes over the former Soviet airbase following the overthrow of the Taleban, using some of the spare hangars for a detention centre.
December 2002
Two prisoners, known only as Dilawar and Habibullah, die in the prison after being tortured. Several US soldiers are later found guilty of abuse but are punished by no more than a few months in prison.
June 2004
British resident Binyam Mohamed is interrogated in Bagram several times - with allegations that he was kept in constant darkness and bombarded with loud music - before being taken to Guantanamo.
July 2005
Four terrorism suspects escape from the prison by picking the locks on their cells and walking out under the cover of darkness.
April 2010
A BBC investigation claims Afghan prisoners have been abused in a "secret jail" at Bagram.
- Independent
