Haji Gulab Mangal, the governor of Afghanistan's Paktika province, said the US military had apologised to the families of the wounded, who he said were in a stable condition.
"Last night (Thursday) the vehicle was driving very fast towards an American checkpoint," Mangal said.
"The Americans wanted to stop the vehicle, but the driver of the vehicle didn't realise and the Americans opened fire on them. Four people were in the car and all of them were wounded."
According to Mangal, the wounded passengers were taken by US troops to a local hospital for treatment.
The incident took place near Orgun-E, a town some 170 km south of Kabul and close to the Pakistan border. The US military was not immediately available for comment.
US forces are stationed in the area as part of their hunt for al-Qaida and Taliban fighters, who are active along the Afghan-Pakistan frontier.
There are around 15,500 American soldiers in Afghanistan. Their prime targets include al-Qaida chief Usama bin Ladin, his deputy Ayman al-Zawahri, the ousted Taliban's supreme leader Mullah Muhammad Umar and renegade warlord Gulbuddin Hekmatyar.