The unknown assailant opened fire at Reina nightclub early on Sunday, before managing to flee amid the chaos.
The motive for the attack is not clear, but suspicion has fallen on the Islamic State group, already linked to at least two terror attacks in Turkey last year.
President Recep Tayyip Erdogan said such groups tried "to create chaos".
"They are trying to... demoralise our people and destabilise our country," he added.
The banned Kurdistan Workers' Party (PKK) has distanced itself from the killings, with the PKK's Murat Karayilan quoted as saying it would "never target innocent civilians".
Earlier, Interior Minister Suleyman Soylu confirmed a "manhunt for the terrorist is under way. Police have launched operations. We hope the attacker will be captured soon."
But as the search continued, the first funerals of those killed at the nightclub were held.
More than half of those killed in the attack - which lasted seven minutes - were foreign, according to the state run Anadolu Agency.
Among them were citizens from Israel, Russia, France, Tunisia, Lebanon, India, Belgium, Jordan and Saudi Arabia.
Details continue to emerge about those killed. Canada's Prime Minister Justin Trudeau has confirmed a Canadian had died in the attack, while a US businessman named as William Jacob Raak, 35, from Delaware was reported to be among the injured.
The club, which sits on the banks of the Bosphorus, is one of Istanbul's most upmarket venues - popular with foreigners and often frequented by singers and sports stars.
However, among the first to die was a security guard, identified as Fatih Cakmak, who had been standing at the door.
His brother said he narrowly escaped a double bombing three weeks ago, having been on duty when Kurdish militants launched an attack near a football stadium, killing at least 44 people, mostly police.
At least 69 people are being treated in hospital, officials said, with three in a serious condition.
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