Huge plumes of smoke blackened the sky
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Three people were injured in the blaze, and an assessment of the damage could only begin once the area had cooled down, officials said.
The authorities say a short circuit may have caused the fire.
But a Kurdish separatist group said it had started it to protest against Turkish "massacres" against the Kurds.
There is no confirmation of the claim sent by the Kurdistan Freedom Falcons (Tak) to pro-Kurdish news agency Firat in the Netherlands.
Ataturk airport, about 16km (10 miles) from central Istanbul, is the main destination for domestic and international flights - and serves as a hub for tourism in Turkey.
Business as usual
The fire broke out at about 1230 GMT and quickly erupted into an inferno of flames and plumes of smoke.
British passenger Paul Collins said the black clouds and flames were twice the height of the building. He initially feared a plane had crashed, he added.
But when he got inside the main passenger terminal - about 1km (0.6 miles) from the cargo area - he said that it seemed to be business as usual.
"In the passenger terminal building, apart from a real feeling of apprehension around, nothing appears to have happened," he told the BBC.
"It is bizarre watching it out the window as people are going about their business.
"It is almost eclipse-like, it is so dark. There are lots of planes coming in and dumping water."
All the workers of the cargo section - up to 2,000 people - made it out safely.
Spark
Occasional explosions were heard as the fire engulfed the cargo building.
Fifty fire engines, planes and firefighters from the air force academy were all scrambled to deal with the emergency.
ATATURK AIRPORT Turkey's largest airport On European side of city Domestic and international Modern international terminal opened in 2000 60,000 passengers a day 15.6 million passengers in 2004
Yellow firefighting planes, usually used to tackle forest fires, roared low over the building every few minutes carrying water to pour on the flames.
Istanbul deputy governor Fikret Kasapoglu says no-one died in the fire and reports suggest just three people have sustained light injuries and smoke inhalation.
Three warehouse buildings have burned to the ground.
"There are various kinds of materials in there, flammable, explosive, so we have to be careful," Mr Kasapoglu said.
But he added: "It was definitely not sabotage."
In its claim, the Tak militants said: "The sabotage is a response to the policies of massacre followed by the Turkish state towards the Kurds.
Turkish authorities have not commented on the claim.
A huge fire that started in the cargo area has engulfed part of Turkey's Atatürk International Airport in Istanbul. All flights to and from the airport have been diverted and major fire-fighting efforts are in progress. 

Hundreds of people have evacuated the area, which is about 1 kilometre away from the passenger terminal.

A witness told Turkish news channel NTV that the blaze broke out in the cargo area of Terminal A, presumably due to an electrical spark, at around 12:30 UTC. The area in which the fire is raging is near the location of the fuel depots. Unconfirmed reports suggest that there were about 250 cargo-handlers present there but most of them seem to have escaped unharmed.

Thirty metre high plumes of smoke are rising into the air and the billows can be seen from all over Istanbul, which indicates the severity of the inferno. Several fire engines and ambulances are rushing to the scene even as aircraft overhead try to douse the fire in an attempt to prevent it from spreading to a hotel nearby. The airport, incidentally, is located very close to a military hangar.