The accident took place late on Sunday when the vehicle was heading toward Faizabad city, the capital of the province, Xinhua news agency reported.
People who live in the area and rescuers rushed to the scene and were attempting to recover the bodies, the statement added.
Last week, eight commuters, including women and children, lost their lives when their vehicle plunged into the same river in the same area.
Road accidents are common in Afghanistan, where traffic rules are rarely followed and roads, especially in rural and mountainous regions, are often in a poor condition.