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At least four other people were hurt when the Costa Europa, carrying 1,473 passengers, came into the harbour during bad weather at 0445 (0145 GMT).
The dead sailors were an Indian, a Honduran and a Brazilian, the General Authority for Red Sea Ports said.
The injured reportedly include three British women and an Italian man.
They were in a stable condition in hospital, officials told the AFP news agency.
In a statement, Costa Cruises confirmed that the Costa Europa had damaged the starboard side of the hull, but said it was "now safely docked in port".
"We sadly have to confirm the deaths of three crew members," the Italian company said.
"The guests have been disembarked and are being accommodated in local hotels," it added.
Witnesses said Sharm al-Sheikh had been hit by fierce winds overnight and that the crash had left a 2m-wide hole in the hull.
The Costa Europa was on an 18-day cruise from Dubai to Savona, and had arrived in Sharm al-Sheikh from the Egyptian port of Safaga.
Comic showing Batman's debut sells for $1m | A comic showing the debut of superhero Batman has been sold for more than $1m (£655,000) at an auction in Dallas. The rare 1939 copy of Detective Comic No 27 was bought by an anonymous bidder from a seller who also wished to keep their identity secret. The sale comes just days after an early edition of a Superman comic sold for $1m - only to be outdone by Batman. Barry Sandoval, of auction house Heritage, claimed it was the biggest price on record for a comic book. "It pretty much blew away all of our expectations," he said. "We can really say that Batman has nosed out Superman, at least for now." Mr Sandoval added that the cover of the comic - which was bought by the seller in the 1960s for $100 (£65) - is one of the most