The International Committee of the Red Cross (ICRC) and the Lebanese Red Cross escorted Khan's remains out of Syria and into Lebanon in a three-car convoy, said an AFP correspondent at the border.
ICRC spokesman Simon Shorno earlier told AFP that Khan's body would be repatriated and taken to Beirut to be handed over to the British embassy.
Previously, Syria's Deputy Foreign Minister Faisal Muqdad had said the ICRC was present at a final autopsy on Khan's body, though Shorno said his organisation "did not take part in any autopsy."
But Syrian authorities said Wednesday that Khan was found "hanging" in his cell, where he was being held for "unauthorised activities," and that he had committed suicide.
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Muqdad said President Bashar al-Assad had decided to grant Khan an amnesty and hand him over to his mother and British MP George Galloway at a news conference in Damascus.
A Syrian monitoring group has also blamed Damascus, saying scores of detainees who had allegedly committed suicide had actually been tortured to death.
Khan, a volunteer with London-based charity Human Aid UK, had travelled to Aleppo in northern Syria last year to help civilians when he was arrested by the regime.