Among the 44 dead are 23 Turkish citizens and at least 10 foreign nationals, three of dual citizenship, Turkish authorities said.
Hundreds of frightened passengers streamed out of the airport, fleeing the latest of several bombings to strike Turkey in recent months. The attacks on a key partner in the US-led coalition against ISIS and a NATO member have increased in scale and frequency. They have scared off tourists and hurt the Turkish economy, which relies heavily on tourism.
Hevin Zini, 12, had just arrived from Duesseldorf, Germany, with her family and was in tears from the shock.
"There was blood on the ground," she told The Associated Press. "Everything was blown up to bits... if we had arrived two minutes earlier, it could have been us."
Yildirim, speaking to reporters at the airport, said all initial indications suggested the Islamic State group was behind the attacks.
"The findings of our security forces point at the Daesh organization as the perpetrators of this terror attack," Yildirim said, using the Arabic name for ISIS. "Even though the indications suggest Daesh, our investigations are continuing."
Turkey shares long, porous borders with Syria and Iraq, war-torn countries where ISIS controls large pockets of territory. Authorities have blamed ISIS for several major bombings over the past year, including on the capital Ankara, as well as attacks on tourists in Istanbul.
Turkey has stepped up controls at airports and land borders and deported thousands of foreign fighters, but has struggled to tackle the threat of ISIS militants while also conducting vast security operations against Kurdish rebels, who have also been blamed for recent deadly attacks.
The devastation at Istanbul's airport follows the March attack on Brussels Airport, where two suicide bombings ripped through check-in counters, killing 16 people. The Islamic State group claimed responsibility for that attack, as well as a subsequent explosion at a Brussels subway station that killed 16 more people.
Yildirim said air traffic at Ataturk Airport, which was suspended after the attack, had resumed to normal early Wednesday. A stoppage of flights to and from the United States and Istanbul lasted several hours but was later lifted, said a US official.
Yildirim said the attackers arrived at the airport in a taxi and blew themselves up after opening fire. Asked whether a fourth attacker might have escaped, he said authorities have no such assessment but are considering every possibility.
Another Turkish official said two of the attackers detonated explosives at the entrance of the international arrivals terminal after police fired at them, while the third blew himself up in the parking lot.
The official cited interior ministry information and said that none of the attackers managed to get past security checks at the terminal's entrance.
Turkish airports have security checks at both the entrance of terminal buildings and then later before entry to departure gates.
Roads around the airport were sealed off for regular traffic after the attack and several ambulances could be seen driving back and forth. Passengers were left sitting on the grass outside the airport.
South African Judy Favish, who spent two days in Istanbul as a layover on her way home from Dublin, had just checked in when she heard an explosion followed by gunfire and a loud bang.
She says she hid under the counter for some time.
Favish says passengers were ushered to a cafeteria at the basement level where they were kept for more than an hour before being allowed outside.
Two South African tourists, Paul and Susie Roos from Cape Town, were at the airport and due to fly home at the time of the explosions.
"We came up from the arrivals to the departures, up the escalator when we heard these shots going off," Paul Roos said. "There was this guy going roaming around, he was dressed in black and he had a hand gun."
The prime minister called for national unity and "global cooperation" in combatting terrorism.
"This (attack) has shown once again that terrorism is a global threat," Yildirim said. "This is a heinous planned attack that targeted innocent people."
He suggested that the attack was linked to what he said was Turkey's success against Kurdish rebels, as well as steps Ankara took Monday toward mending strained ties with Israel and Russia.
"It is meaningful that this heinous attack came at a time when we have become successful in the fight against separatist terrorism ... and at a time when we started a process of normalizing ties with our neighbors," Yildirim said.
Yildirim said there was no security lapse at the airport, but added the fact the attackers were carrying weapons "increased the severity" of the attack.
The victims included some foreigners, he said, adding that many of the wounded have minor injuries but others were more badly hurt.
Saudi Arabia's Embassy in Turkey said at least seven Saudis were injured in the attack and all are in stable condition
The independent DHA news agency said the wounded, among them police officers, were transferred to Bakirkoy State Hospital.
Turkey is beset by a wide array of security threats, including from ultra-left radicals, Kurdish rebels demanding greater autonomy in the restive southeast, and ISIS militants.
On Jan. 12, an attack that Turkish authorities blamed on ISIS claimed the lives of a dozen German tourists visiting Istanbul's historic sites. On March 19, a suicide bombing rocked Istanbul's main pedestrian street, killing five people, including the bomber, whom the authorities identified as a Turkish national linked to ISIS.
Last October, twin suicide bombings hit a peace rally outside Ankara's train station, killing 102 people. There was no claim of responsibility but Turkish authorities blamed the attack on a local cell of ISIS.
Istanbul's Ataturk Airport was the 11th busiest airport in the world last year, with 61.8 million passengers, according to Airports Council International. It is also one of the fastest-growing airports in the world, seeing 9.2 percent more passengers last year than in 2014.
The largest carrier at the airport is Turkish Airlines, which operates a major hub there. Low-cost Turkish carrier Onur Air is the second-largest airline there.
The independent Dogan news agency reported that a plane carrying Albanian Prime Minister Edi Rama was arriving on an official visit at the airport when the attack occurred. The prime minister and his entourage were safely taken to an official residence.