The High Court ruled Hamza, Babar Ahmad, Syed Talha Ahsan, Adel Abdul Bary and Khaled al-Fawwaz did not show "new and compelling" reasons to stay.
The European Court of Human Rights had previously backed successive UK court rulings that they could be extradited.
The men left Long Lartin prison in Worcestershire in a police convoy.
Three police 4x4s, two armoured vans and a blacked-out police people carrier left the jail at 19:15 BST.
The men are travelling with members of the Metropolitan Police's extradition unit to RAF Mildenhall in Suffolk where they will be handed over to US marshals.
The BBC understands two US civilian jets - one of which is registered to the US Department of Justice - are on the tarmac at the airbase.
A Home Office spokesman welcomed the decision and said it was "working to extradite these men as quickly as possible".
The US first attempted to extradite Abu Hamza in 2004 over 11 allegations, including of a conspiracy to take hostages and hostage-taking in Yemen in 1998, in an incident that led to the death of four people.
The process was halted when the UK decided to try him on allegations of soliciting to murder and stirring up racial hatred relating to his sermons. He was convicted in 2006.
Judges Sir John Thomas and Mr Justice Ousley said in their ruling that there was an "overwhelming public interest in the functioning of the extradition system" and that there was "no appeal from our decision".
Of the long legal battle to send the men - whose extradition requests were submitted between 1998 and 2006 - to the US, Sir John told the court: "It is unacceptable that extradition proceedings should take more than a relatively short time, to be measured in months not years.
"It is not just to anyone that proceedings such as these should last between 14 and eight years."
As one legal process was coming to a close in the UK, another was getting under way in the US.
The five suspects will enter US criminal justice system. It is expected they will all face a court hearing in the next few days during which they will hear the charges against them.
For Abu Hamza al Mazri, Khalid al Fawaz and Adel Abdul Bary, this will happen in New York.
But Babar Ahmed and Talhar Ahsan, it is thought, will have their hearings in Connecticut, where the courts have jurisdiction over their case. Judges will set dates for their trials.
Officials in the Department of Justice expressed relief that the extradition process was finally over, and exasperation at its duration.
They would not comment on the conditions in which the suspects will be held, but it's likely they will go to maximum security facilities, as would anyone charged with crimes related to terrorism. The officials said there was no question of any of the men facing the death penalty, or entering the military justice system at Guantanamo Bay.
The extraditions have received little attention in the US. The name Abu Hamza does not resonate here, and it is unlikely his trial will mean much to an American public wearied by years of terrorism cases.
There was no doubt each man had, over the years, "either taken or had the opportunity to take every conceivable point to prevent his extradition to the United States", he added.
The judges' written ruling, read out in court, concluded that "each of the claimants' applications for permission to apply for judicial review or for a re-opening of the statutory appeals be dismissed".
The judges rejected a plea by Abu Hamza to delay his extradition so he could undergo an MRI brain scan which, his lawyers said, could show he was unfit to plead because of degenerative problems.
"The sooner he is put on trial the better," they said.
The 54-year-old, a former imam at Finsbury Park mosque, north London, was suffering from chronic sleep deprivation and depression as a result of eight years in prison, his lawyers added.
But during the hearing, Sir John observed there were "excellent medical facilities in the United States".
The judges also rejected appeals from Mr al-Fawwaz and Mr Bary, who are accused of being aides to Osama Bin Laden in London.
The US alleges they promoted violent jihad against the West and were involved in the 1998 US embassy bombings in East Africa, which killed more than 200 people.
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The battle to stay in the UK is also over for Babar Ahmad and Talha Ahsan, who are accused of running pro-jihad website Azzam.com - which the US says was hosted there - and of helping terrorists.
In a statement prepared in prison before the verdict, Mr Ahmad said that, "By exposing the fallacy of the UK's extradition arrangements with the US, I leave with my head held high having won the moral victory."
In a statement released by its embassy in London, the US said it was "pleased" the men were being extradited after "a lengthy process of litigation".
"The law enforcement relationship between the United States and United Kingdom is predicated on trust, respect, and the common goals of protecting our nations and eliminating safe havens for criminals, including terrorists," it added.
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