They said that the unprecedented salvage effort "reached degree zero (vertical), which was our target".
In the operation that took all of Monday and most of the night, they used cables and metal boxes filled with water to roll the ship onto a platform.
The Costa Concordia capsized in January 2012, killing 32 people.
The bodies of two of the victims of the disaster, by the island of Giglio, have never been found. There are hopes that they may be located during the operation.
The ship was declared completely upright shortly after 04:00 local time (02:00) on Tuesday.
Franco Gabrielli, the head of Italy's Civil Protection Authority, said the vessel was now sitting on a platform built on the sea bed.
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