A helicopter gunship was called in to support the three vehicle patrol.
One witness said: "The helicopters landed, the soldiers came forward on foot. The people in the cars got out and waved their hands above their heads. But they still opened fire.
"It was a very bad situation."
However, the US military said that the men were suspected opposition fighters who had shot at them first.
US and Afghan authorities, who have launched a joint investigation into the incident, said that the shots caused many secondary explosions.
Karzai said in the statement that the men who died were all employees of a private security company to protect a construction firm who were building a road in the province.
Arsala Jamal, the Khost governor, said: "None of them is alive to say how it happened ... but I know they were not Taliban.
"They were security guards working for $250 a month."
The US military statement said that their forces responded to fire from three vehicles carrying the men.
It said the men "got out of their vehicles and fired on the coalition forces, who returned fire with rifles and helicopter gun fire.
"There were secondary explosions in the vehicles, and 14 armed men were killed. Numerous ammunition belts and small arms weapons were recovered from the vehicles."
Al Jazeera's David Chater, reporting from the capital Kabul, said that the Taliban and al-Qaeda are both active in the Khost area.
It is the second such attack in a month, with local officials saying that 23 security guards were killed by a US fighter jet in October. The guards had set up roadblocks to protect a construction team in Ghazni, central Afghanistan.
There have been scores of claims of Afghan civilian deaths at the hands of international soldiers, tens of thousands of which are in the country to fight Taliban fighters.
Last week, Karzai urged Barack Obama, the US president-elect, to ensure that US-led operations do not lead to civilian deaths.