The mishap took place as a the driver lost control over the vehicle and fell into a roadside ditch, Jhanjharpur police station in-charge Pawan Kumar said.
Two persons, identified as Abdul Sattar (55) and Raju Ram (45) died on the spot, while nine others sustained injuries, he said.
The injured were admitted to Jhanjharpur Primary Health Centre (PHC) for treatment, while the bodies have been sent to the Sadar hospital for autopsy, Kumar said.
The bus was carrying passengers from Jhanjharpur to Madhupur in the district, the police officer said.
