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US Secretary of State Antony Blinken has said the Taliban have promised to allow foreigners and Afghans to leave Afghanistan beyond 31 August, when a US-controlled airlift will end.
He said Washington and its allies had "a responsibility to hold [the Taliban] to that commitment".
Mr Blinken said the US was still on track to complete operations at Kabul airport by the end of the month.
There are still thousands inside and outside the airport wanting to fly out.
More than 82,000 have been airlifted from Kabul, which fell to the Taliban 10 days ago. The militants have opposed extending the deadline.
The UK and other countries are also rushing to evacuate their nationals, as well as Afghan staff and visa holders, ahead of the deadline. The UK said its evacuation operation was moving at "significant pace", with some 1,200 people flown out on Wednesday .
About 19,000 people have been evacuated on US-organised flights in the past 24 hours, Mr Blinken said, with airlifts stepped up in recent days amid scenes of chaos.
"Only the United States could organise and execute a mission of this scale and this complexity," he told reporters in Washington.
"The Taliban have made public and private commitments to provide and permit safe passage for Americans, for third-country nationals and Afghans at risk past August 31st," he said.
He added that the US would help those who wanted to leave Afghanistan "not just during the duration of our evacuation and relocation mission, but every day thereafter".
Up to 1,500 US nationals could still be in the country and the administration was making exhaustive efforts to trace them, the secretary of state said.
The CIA and the US military have been carrying out clandestine operations to extract Americans using helicopters and ground troops, US and other officials have told the Wall Street Journal.
The perilous operations have been both inside and outside of Kabul, the paper says.
The Pentagon has said 10,000 people are still waiting to be evacuated from Kabul airport by US aircraft. There are fears for thousands more Afghans who are desperate to leave but cannot reach the site.
BBC correspondents say many of those being turned back by the Taliban at the airport gates appear to have travel papers.
President Joe Biden tweeted that the US would "do everything" to "provide safe evacuation".
The BBC is not responsible for the content of external sites. View original tweet on TwitterUK Foreign Secretary Dominic Raab said his government would use "every hour and day that we've got left" to fly British nationals and eligible Afghans out.
President Biden has rejected calls from allies to extend the 31 August deadline. US officials have said there is a significant threat that militants linked to the Islamic State group could target the airport.
On Wednesday, the UK Foreign Office urged its nationals not to travel there because of the risk of a terror attack .
Kabul airport is currently being defended by 5,800 US and 1,000 British troops.
Turkey, for its part, announced that it had started pulling its troops out of Afghanistan.
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