The assailant shot dead a policeman and a civilian at the entrance to the Reina club, one of the city's most exclusive nightspots, and then went on a shooting rampage inside, Turkish officials said.
Interior Minister Suleyman Soylu said the attacker escaped and was now the target of a major manhunt, expressing hope the suspect "would be captured soon".
Soylu said in televised comments that of 21 victims who have been identified so far, 16 are foreigners and five are Turks. Another 69 people are being treated in hospital.
Many revellers threw themselves into the water in panic and efforts were under way to rescue them, NTV television said.
Dogan news agency said there were two gunmen dressed in Santa Claus outfits, although this has yet to be confirmed.
Television pictures showed party-goers - including men in suits and women in cocktail dresses - emerging from the nightclub in a state of shock.
"What happened today is a terror attack," he said.
Dogan reported that some witnesses claimed the assailants were "speaking Arabic" while NTV said special force police officers were still searching the club.
There has been no claim of responsibility.
The attack evoked memories of the November 2015 carnage in Paris when Islamic State jihadists went on a gun and bombing rampage on nightspots in the French capital, killing 130 people including 90 at the Bataclan concert hall.
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