Local television 1TV reported that multiple explosions were heard in the capital after dark, but the city was largely quiet during the day on Sunday.
Aid group Emergency said 80 wounded people had been brought to its hospital in Kabul, which was at capacity, and that it was only admitting people with life-threatening injuries.
In a Facebook post, Ghani said he had left the country to avoid clashes with the Taliban that would endanger millions of residents of Kabul.
He did not say where he was and it was not clear where he was headed or how exactly power would be transferred following the Taliban's lightning sweep across Afghanistan.
Al Jazeera earlier showed footage of what it said were Taliban commanders in the presidential palace with dozens of armed fighters.
Some local social media users in Kabul branded Ghani a coward for leaving them in chaos. A tweet from the verified account of the Afghan Embassy in India said: "We are all banging our heads in shame."
Many Afghans fear the Taliban will return to past harsh practices in their imposition of sharia, or Islamic religious law. During their 1996 to 2001 rule, women could not work and punishments such as stoning, whipping and hanging were administered.
The militants sought to project a more moderate face, promising to respect women's rights and protect both foreigners and Afghans.
"We are ready to have a dialogue with all Afghan figures and will guarantee them the necessary protection," Naeem told Al Jazeera Mubasher TV.
United Nations Secretary-General Antonio Guterres urged the Taliban and all other parties to exercise the utmost restraint, and expressed particular concern about the future of women and girls in Afghanistan.
The Pentagon authorized another 1,000 troops to help evacuate US citizens and Afghans who worked for them, a US official said.
A senior US defense official told Reuters on Sunday evening in Washington that about 500 people, mostly Americans, had so far been evacuated, and that the number would rise to 5,000 a day when all planned US forces are in Kabul.
European nations, including France, Germany and the Netherlands, also said they were working to get citizens as well as some Afghan employees out of the country.
Russia said it saw no need to evacuate its embassy for the time being. Turkey said its embassy would continue operations.
Asked if images of helicopters ferrying personnel were evocative of the United States’ departure from Vietnam in 1975, US Secretary of State Antony Blinken told ABC News: "Let's take a step back. This is manifestly not Saigon."
Biden has faced rising domestic criticism after sticking to a plan, initiated by his Republican predecessor, Donald Trump, to end the US military mission in Afghanistan by Aug 31.
In a statement on Sunday, Senate Republican leader Mitch McConnell blamed Biden for what he called a "shameful failure of American leadership".
"Terrorists and major competitors like China are watching the embarrassment of a superpower laid low," McConnell said.
Naeem said the Taliban would adopt a policy of non-interference in others' affairs in return for non-interference in Afghanistan.
"We do not think that foreign forces will repeat their failed experience in Afghanistan once again."