Mr Biden called it one of the largest and most difficult airlifts in the history of the United States.
“Let me be clear: any American who wants to come home, we will get you home,” he said.
The president said the number included not just American citizens and permanent residents, but also their families; as well as applicants for special immigration visas like translators and interpreters and their families; women leaders; and journalists.
“We're going to do everything, everything that we can to provide safe evacuation for our Afghan allies, partners, and Afghans who might be targeted because of their association with the United States,” he said.
And he added: “I cannot promise what the final outcome will be ... or that it will be without risk of loss.
“But as commander in chief, I can promise you I will mobilise every resource necessary, and as an American, I offer my gratitude to the brave men and women of the US armed forces who are carrying out this mission. They’re incredible.”
Mr Biden also cancelled a planned trip to his home in Delaware on Friday to deal with the situation in Kabul.
He had been due to leave the White House but remained in Washington DC on Friday night following his speech, although it is unknown if he will still head home on Saturday.
During his address the president said thanks to coordination with The New York Times, The Wall Street Journal and The Washington Post, all 204 of their employees were evacuated.
“We’ve established the flow of flights and we’ve increased the number of people moving out of the country,” he said.
But when Mr Biden was asked whether he sent US troops into Kabul to evacuate Americans who haven’t been able to get to Hamid Karzai International Airport safely, he said there was no indication that anyone has been unable to get to Kabul through the airport.
“We’ve made an agreement with the Taliban thus far, they’ve allowed them to go through, it’s in their interest for them to go through,” he said. “So we know of no circumstance where American citizens carrying an American passport are trying to get through to the airport, but we will do whatever needs to be done to see to it that they get to the airport.”
Mr Biden addressed the nation and faced questioning after days of scathing criticism of the chaotic evacuation that has unfolded at Kabul’s international airport.
Earlier in the week, the world watched on horrified as thousands of desperate Afghans stormed onto the runway to try and climb onboard flights leaving the country.
The US Air Force said that it had found human remains inside the wheel well of a C-17 cargo plane, and video captured Afghans falling from the plane as it took off.
Mr Biden on Wednesday told ABC News that there was no way that the US could have left the country after 20 years without “chaos ensuing.”
And flights out of Kabul were paused for eight hours earlier on Friday as the US tried to find places for them to go, after a processing facility in Qatar reached capacity.
Mr Biden told reporters that the US had evacuated a total of 18,000 people since July.
“We have already secured a number of agreements for these passengers to temporarily transit through other countries, and have been working aggressively to secure additional agreements,” a senior White House official said after Mr Biden’s address.
“We are deeply grateful for the generosity of our international allies and partners, including those who are working shoulder-to-shoulder with us on the ground in Kabul to support what is already one of the largest airlifts in history.”
The White House has defended itself from the withering criticism that followed the collapse of the Afghan army and US-backed government as the withdrawal of American troops got underway.
Amid all of the chaos there was positive news, as a baby handed up over an airport fence to a US Marine was reunited with its father and was in good health.
“The baby seen in the video was taken to a medical treatment facility on site and cared for by medical professionals,” said Marine Corps spokesperson, Major Jim Stenger.
Register for free to continue reading
Registration is a free and easy way to support our truly independent journalism
By registering, you will also enjoy limited access to Premium articles, exclusive newsletters, commenting, and virtual events with our leading journalists
Already have an account? sign in
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies