The 290-metre, 114,500-ton ship has lain on its side ever since foundering off the Tuscan coast in January 2012 in a tragedy that claimed 32 lives.
The unprecedented operation, which was delayed by several hours by storms, began after a maritime exclusion zone was established around the site.
"The senior salvage master gave the order to activate the commands," the civil protection agency, which is overseeing the salvage, said in a statement.
Salvage coordinators say the lifting could last up to 12 hours and warn there will be some spillage of the ship's waste into the pristine Mediterranean waters.
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"The concentrations will be limited. There is no contamination problem," said Giandomenico Ardizzone, a marine biology professor working on the project.
Ardizzone said 29,000 tons of waste including heavy metals and fuel will pour into the sea but the quantity of toxic waste will not be sufficient to cause permanent damage.
The bigger concern for the salvage workers is whether the weakened hull of the ship can withstand the enormous pressure it will be under as it is winched up.
The man giving the orders from a control room on a barge next to the Costa Concordia is Nick Sloane, a South African salvage master with years of experience on spectacular shipwrecks around the world.
Sloane and his team of experts monitored progress on eight monitors and five cameras and five microphones have been placed on the top deck of the Concordia.
Islanders whose lives have been turned upside-down by the wreck said they were relieved that the time when the ship will finally be removed was drawing closer.
Even if the ship is lifted upright, they will have months more to wait, as the towing away is not planned until spring of next year after winter storms.
Special prayers were held in a local church on the eve of the operation yesterday for the victims of the wreck and for the success of the salvage.
"The sooner it happens, the better," said the parish priest, Father Lorenzo Pasquotti, who opened his church to survivors on the night of the disaster.
The Costa Concordia was once a floating pleasure palace filled with entertainment and sporting facilities, including four swimming pools and the largest spa centre ever built on a ship.
It struck a group of rocks off Giglio after veering sharply towards the island in a bravado sail-by allegedly ordered by captain Francesco Schettino.