Three attackers arrived in a taxi and began firing at the terminal entrance late on Tuesday. They blew themselves up after police fired back.
PM Binali Yildirim said early signs pointed to so-called Islamic State.
However, no-one has so far admitted carrying out the attack.
Turkey has declared Wednesday a day of national mourning.
Turkish investigators are examining CCTV footage, witness statements and mobile phone video recorded by terrified passengers to try to determine the identity of the attackers.
The Dogan news agency said autopsies on the three dead men suggested they may be foreign nationals but this has not been confirmed.
Istanbul’s governor confirmed that 41 people had died, at least 13 of them of foreign or dual nationality.
Some 109 of those injured have now been discharged from hospital.
Cleaners worked through the morning to sweep up shattered glass, while workers repaired cables and ceiling tiles. Heavily-armed security personnel were patrolling the airport.
Flights had resumed in the early morning, though with many cancellations and delays.
Turkish President Recep Tayyip Erdogan said the attack should serve as a turning point in the global fight against militant groups.
Reports of the attack vary but it appears the attackers opened fire at the entrance where X-ray machines are positioned, sparking an exchange with police. At least two of the attackers ran into the building.
Footage on social media shows one moving through the building as people around him flee. He is shot by police and remains on the ground for about 20 seconds before blowing himself up. All three attackers were killed.
Paul Roos, who was due to fly home to South Africa, told Reuters he saw one of the attackers.
“He was wearing all black. His face was not masked. We ducked behind a counter but I stood up and watched him. Two explosions went off shortly after one another. By that time he had stopped shooting.
“He turned around and started coming towards us. He was holding his gun inside his jacket. He looked around anxiously to see if anyone was going to stop him and then went down the escalator. We heard some more gunfire and then another explosion, and then it was over.”
The US called the attack “heinous”, saying America remained “steadfast in our support for Turkey”.
Russian President Vladimir Putin offered his condolences to Turkey.
An Entrepreneur and Philanthropist. Editor-in-Chief of www.233times.com. A Senior Journalist with Ghanaian Chronicle Newspaper. An alumnus of Adisadel College where he read General Arts. He holds first degree in Bachelor of Arts from the University of Ghana; Political Science (major) and History (minor). He has also pursued MSc Corporate Social Responsibility (CSR) and Energy with Public Relations (PR) at the Robert Gordon University in the United Kingdom. He is a 2018 Mandela Washington Fellow (YALI) who studied at Clark Atlanta University on the Business and Entrepreneurship track. His mentors are Rupert Murdoch, Warren Buffet, Sam Jonah, Kwaku Sakyi Addo and Piers Morganview all posts by: Nana Kwesi Coomson
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