The European Court of Human Rights has backed the extradition of Abu Hamza and four other terror suspects from the UK to the US.
The Strasbourg court held there would be no violation of human rights for those facing life and solitary confinement in a "supermax" prison.
Judges said they would consider further the case of another suspect because of mental health issues.
Prime Minister David Cameron said he was "very pleased" with the news.
"It's quite right that we have a proper legal process, although sometimes you can be frustrated by how long things take," he added.
The court's decision is one of its most important since 9/11 because it approves of human rights in US maximum security prisons, making it easier for the UK to send suspects to its closest ally.
There could still hypothetically be an appeal against the court's ruling in its final Grand Chamber - but in practice, very few cases are re-examined in that final forum.
The men have three months to try to persuade the Grand Chamber to reopen the entire case and examine it. If the men fail to launch an appeal, they will be extradited to the United States.
The family of one of the men, Babar Ahmad, who has been held for a record of nearly eight years without trial, said he would fight on against extradition.
Last week, he appealed in a BBC interview to be charged and tried in the UK because his alleged crimes were committed here.
Home Secretary Theresa May welcomed the ruling, and said she would work to ensure that the suspects were handed over to the US authorities "as quickly as possible".
The US Justice Department also said it was "pleased" about the decision on the five.
"We look forward to the court's decision becoming final and to the extradition of these defendants to stand trial in the United States," it said in a statement.
In the case of the sixth suspect, Haroon Aswat, it said officials would "consult" with the UK's Home Office about the additional submission requested.
The European Court said there would be no breach of human rights if the men were to be held in solitary confinement at ADX Florence, a Federal Supermax jail in Colorado, used for people convicted of terrorism offences.
Abu Hamza is unlikely to be held at that jail because of his disabilities. The court also held that the life sentences each man faces would not breach human rights.
But in Mr Aswat's case, judges said they could not yet give the go-ahead to extradition because they needed to see more submissions on his schizophrenia and how that would be treated were he sent to the US.
The court said that the range of activities and services at ADX Florence was better than that at many European prisons.
It said: "Having fully considered all the evidence from both parties, including specifically prepared statements by officials at ADX Florence as well as letters provided by the US Department of Justice, the court held that conditions at ADX would not amount to ill-treatment.
"As concerned ADX's restrictive conditions and lack of human contact, the court found that, if the applicants were convicted as charged, the US authorities would be justified in considering them a significant security risk and in imposing strict limitations on their ability to communicate with the outside world.
"The court finds that there are adequate opportunities for interaction between inmates. While inmates are in their cells talking to other inmates is possible, admittedly only through the ventilation system.
"Save for cases involving the death penalty, it has even more rarely found that there would be a violation of Article 3 (that no-one shall be subjected to torture or to inhuman or degrading treatment or punishment) if an applicant were to be removed to a state which had a long history of respect of democracy, human rights and the rule of law."
Abu Hamza is charged with offences relating to hostage taking in Yemen and an alleged plot to set-up a terrorism training camp in the United States. Haroon Aswat is also accused in connection to the training camp.
Babar Ahmad and Talha Ahsan are accused of supporting terrorism through a website operated in London.
The final two men, Adel Abdul Bary and Khaled al-Fawwaz, allegedly played a part in organising the 1998 US Embassy bombings in East Africa.
Responding to the ruling, the law firm representing Babar Ahmad and others in the case - Birnberg Peirce and Partners - said the court had made its decision "in large part on the basis of disputed statistics provided by the UK government to which the applicants were not permitted to respond".
It also stressed that the judgement did not address the "burning issue" of "why in all logic, fairness, and practical common sense are not British citizens (whose UK actions are forming the basis of prosecution in the US, and where all of the evidence on which they are being tried was accumulated in its entirety in the UK by UK police and shipped lock stock and barrel to US prosecutors), being tried in their own country?"
In an unrelated case earlier this year, the European Court blocked the deportation from the UK of a different radical cleric, Abu Qatada, to Jordan, saying he faced an unfair trial.
Shadow home secretary Yvette Cooper welcomed the ruling against Abu Hamza, but added: "The government now needs to focus on dealing with Abu Qatada, who could have less than a month left of his strict bail conditions, and where the government's own decision to water down counter-terror powers could mean he is allowed to move around London."
Lib Dem MP Julian Huppert - who sits on the Home Affairs Select Committee - acknowledged that the US had "a proper legal system", but said the case flagged up "genuine concerns" over extradition.
He called for authorities to "be more active about charging people who are accused of terrorist activities here in the UK". More needed to be done, he added, to make the system "fast and prompt".