Officials said they expected the 114,500-ton vessel, which has lain on its side since the deadly crash on January 13, 2012, to be upright by 0200 GMT on Tuesday.
The 290-metre (951-foot) long ship -- longer than the Titanic and more than twice as heavy -- could be seen emerging from the water like a ghost ship.
The side of the ship that had been underwater was rusty and brown after 20 months in the sea, contrasting with the white of the exposed side.
It was being carried out with 36 giant cables across the hull and tanks the size of 11-storey buildings welded on the side of the ship which were being gradually filled with water to act as ballast.
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The project has cost 600 million euros (USD 800 million) and insurers, who are picking up the bill, estimate it could run to USD 1.1 billion once it is completed.
The 14-deck Costa Concordia was once a floating pleasure palace with a casino, four swimming pools and the largest spa centre ever built on a ship.
Dubbed "Captain Coward" and "Italy's most hated man" for apparently abandoning the ship while passengers were still on board, Schettino is currently on trial.
Four crew members and the head of ship owner Costa Crociere's crisis unit have already received short prison sentences for their roles in the crash.
The ship had 4,229 people from 70 countries on board.
It keeled over in shallow waters within sight of Giglio's port but the order to abandon the vessel came more than an hour later -- a fatal delay.
Two bodies -- that of an Indian waiter and an Italian passenger -- were never recovered from the wreck and are believed to be still stuck under the ship.
Kevin Rebello, the waiter's brother, and Elio Vincenzi, the passenger's husband, were expected to arrive on Giglio later on Tuesday as prosecutors were expected to launch a new search for the bodies.
"I am still hoping to find my wife. This is a tense wait for me and for my daughter," Vincenzi said.