Police officers and medics are on board the flight
The C-130 Hercules transport aircraft left Ben Gurion airport in Tel Aviv, transporting the men involved in the five-week siege at the Church of the Nativity in Bethlehem.
The men - described by Israel as most-wanted terrorists - are to be given temporary sanctuary in Cyprus until their final place of exile is decided.
Earlier the aircraft had left RAF Akrotiri, in Cyprus, with a number of British military police officers and medics on board.
A Foreign Office spokesman refused to give any further details or outline the involvement of UK diplomatic efforts in bringing to an end the siege.
But he said 'Britain stood ready to help'.
Cypriot foreign minister Ioannis Cassoulides said the men to Cyprus's Larnaca airport where they will be handed over to Cypriot authorities.
It is a temporary host country ahead of their dispersal among several countries, including Italy and Spain.
Cyprus said it would take temporary charge of the 13, to clear the way for an end to the siege.
The 13 were to be followed by 26 who are to be taken to Gaza and about 85 civilians, who are to be freed.
An Italian foreign ministry official yesterday said that under the latest proposed deal, Italy and Spain would take some of the militants, while Austria, Greece, Luxembourg, Ireland and perhaps even Canada might take the rest.
Details of the men's final exiles will worked out at a meeting of EU foreign ministers in Brussels on Monday.