Police said the bus rolled about 100 meters (320 feet) before it plunged into the river near Lamachaur village, about 400 kilometers (250 miles) east of the capital, Katmandu.
Soldiers and police helped by villagers pulled the dead and injured from the wreckage. The injured were transported to nearby hospitals for treatment.
Police said the cause of the accident was being investigated, but that the bus appeared to be overcrowded with pilgrims.
Accidents in Nepal are often blamed on poorly maintained vehicles and roads.