The hulking vessel about twice the size of the Titanic was towed into the northern port after a four-day, 280 kilometre journey from the disaster site off the Tuscan island of Giglio.
"We can finally breathe a sigh of relief," Italy's Environment Minister Gian Luca Galletti told journalists.
Fears the damaged hull would break up under the strain, spilling toxic waste into Europe's biggest marine sanctuary, proved unfounded, and dolphins joined the convoy of environmental experts in welcoming the ship into Genoa.
Civil protection agency chief Franco Gabrielli told journalists a high wind was slowing the delicate operation and the ship was not expected to be secured until around 1400 GMT.
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Once it is fastened in place, interior furnishings and fittings will be stripped out of the ship to make it light enough to tow into the scrapping area, where it will be divided into three parts for dismantling.
Crowds of curious locals gathered near the port on the outskirts of Genoa at first light, eager to see the remains of the battered ship, which crashed into rocks off Giglio island in January 2012 with 4,229 people from 70 countries on board.
At a floating ceremony in 2006 attended by Cardinal Tarcisio Bertone -- Pope Benedict XVI's number two -- the bottle of champagne swung against the hull failed to smash, a bad omen in seagoing lore.
Images of the vast vessel toppled on its side off Giglio in January 2012 went viral around the world, and its captain Francesco Schettino was dubbed Italy's "most hated man" by local media after he escaped in a lifeboat while terrified passengers threw themselves into the icy sea.
The salvage operation to recover the Concordia was the biggest ever attempted and is expected to cost in the region of USD 2.01 billion.