The court ruled Hamza, Babar Ahmad, Syed Talha Ahsan, Adel Abdul Bary and Khaled al-Fawwaz did not show "new and compelling" reasons to stay in the UK.
Their appeal came after the European Court of Human Rights backed successive UK courts in ruling for extradition.
The men have left Long Lartin prison in a police convoy.
Three police 4x4s, two armoured vans and a blacked-out police people carrier arrived at the Worcestershire jail at 18:15 BST.
A Home Office spokesman welcomed the decision and said it was "working to extradite these men as quickly as possible".