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The US President addressed the nation after it emerged that 13 US service personnel were among 90 people killed after two blasts and a gunfight outside the airport on Thursday.
Some 143 people, including 15 US personnel, were injured in the attack. Islamic State claimed responsibility.
President Biden blamed an affiliate of the so-called Islamic State in Afghanistan, Isis-K, for the attack on Kabul airport.
In a press conference, he said the US intelligence community has assessed that the attack was carried out by Isis-K.
He said: "The intelligence community has assessed... an attack by a group known as Isis-K took the lives of American service members standing guard at the airport and wounded several others seriously."
He added: "With regard to finding, tracking down, the Isis leaders who ordered this, we have some reason to believe we know who they are, we are not certain, and we will find ways of our choosing, without large military operations, to get them wherever they are."
Mr Biden said the Americans killed in attacks were "heroes" who were "engaged in a dangerous, selfless mission to save the lives of others”.
The president vowed to complete the evacuation of American citizens and others from Afghanistan despite the deadly suicide bomb attack.
As many as 1,000 Americans and many more Afghans are still struggling to get out of Kabul.
Mr Biden was briefed on the attacks, which came 12 days into the rushed evacuation and five days before its scheduled completion. Some Republicans argued to extend the evacuation beyond next Tuesday’s deadline.
The US general overseeing the evacuation, Gen Frank McKenzie, said after the attacks: “If we can find who is associated with this, we will go after them.”
He said it would be a mistake for the United States to call an early end to the evacuation, despite the risks.
The administration has been widely blamed for a chaotic and deadly evacuation that began in earnest only after the collapse of the US-backed Afghan government and the Taliban’s takeover of the country.
More than 100,000 people have been evacuated so far.
Thursday’s attacks were sure to intensify political pressure from all sides on Mr Biden, who already was under heavy criticism for not beginning the pullout earlier.
He had announced in April that he was ending the US war and would have all forces out by September.
House Republican leader Kevin McCarthy called for Speaker Nancy Pelosi to bring the chamber back into session to consider legislation that would prohibit the US withdrawal until all Americans are out.
That is highly unlikely, and Ms Pelosi’s office dismissed such suggestions as “empty stunts”.
At the Pentagon, Gen McKenzie said the military believes the attacks on the airport’s perimeter were carried out by fighters affiliated with the Islamic State group’s Afghanistan arm. He said more attempted attacks were expected.
After the suicide bomber’s attack at the airport’s Abbey Gate, a number of Isis gunmen opened fire on civilians and military forces, he said. There also was an attack at or near the Baron Hotel near that gate, he said.
Mr Biden had pledged to get out of Afghanistan every American who wished to leave. As of Thursday, the State Department estimated there were as many as 1,000 Americans in Afghanistan who may want help getting out.
Watch: '‘She died right in my arms’: Afghan witness says small child was victim of Kabul blasts