Lothian Buses revealed the bill over the last three years to the Evening News under freedom of information laws but insisted the 2045 accidents represented a tiny percentage of the total number of journeys.
The most common complaint filed against the council-owned company is for injuries suffered while travelling inside the vehicle, with an average of 398 cases reported each year.
On average 98 people claim compensation after falling while boarding or leaving, while there have been 157 cases since 2007 of passengers getting caught in the bus doors.
The city council's transport leader Gordon Mackenzie described the accidents as "regrettable" but said he was reassured by the company's safety and customer service record.
"The fact that a large proportion of the passengers that they carry are elderly means they have to be on top of driver training," he said.
"Obviously there are occasions when buses have to stop or start abruptly, (but] there's also a very good system in place for picking up on drivers who are not behaving as the public feel they should."
Opposition councillor Allan Jackson, the Tories' transport spokesman, said: "Accidents are always a possibility in a city like Edinburgh. Although Lothian Bus drivers are very good, we can't expect it to run smoothly forever."
Campaign officer for Age Scotland, Doug Anthoney, said accessible public transport was vital for the many older people who do not have access to a private car: "We would ask all bus companies to think about what more they can do to assist and reassure their older customers with regard to safety."
Lothian Buses said the number of accidents had been declining. A spokesman said: "With a business which last year carried well over 100 million passengers there are bound to be a number of accidents. We are never complacent but we have had some success in recent years in moving that number downwards."