Officials said the toll could rise as most of the injured have suffered grievous injuries.
The bus carrying around 40 passengers was on its way from Kalpa to Sangla.
It met with the accident near Sangla, some 300 km from here, Deputy Commissioner D.D. Sharma told IANS from the spot.
He said a request had been sent to the authorities in Shimla to send a helicopter to airlift the injured to the Indira Gandhi Medical College and Hospital in Shimla.
The cause of the accident is yet to be ascertained, he said over phone from the spot.
The injured have been admitted to nearby government hospitals.
Witnesses said police had a tough time extricating the bodies from the mangled remains of the bus.
Chief Minister Virbhadra Singh expressed shock and grief over the accident and directed the district administration to take all possible measures to rescue the victims and provide them adequate relief.
Kinnaur is one of the state's remotest districts.