Michael Turner, 47, suffered head wounds, concussion, and neck and shoulder injuries in the assault at HMP Woodhill in Milton Keynes.
Bronson, 61, whose history of violence earned him public notoriety, is serving a life sentence for robbery and kidnap.
He attacked Mr Turner, a dog handler, outside the prison's gym in April 2010.
High Court judge John Mitchell said Bronson was "was smashing up equipment within the prison gym" and had been locked in by staff who were trying to contain the situation.
Bronson used gym equipment to smash through the door and started attacking Mr Turner and his German Shepherd dog who were stationed outside.
"He punched Mr Turner around his head with his fists, causing him to fall to the ground," said the judge.
"The prisoner then continued to hit him and jumped on him - pushing Mr Turner, and his dog who was under him, on to the floor with his full body weight."
Prison officers managed to pull Bronson away from Mr Turner, who later sued the Ministry of Justice for failing to provide support when dealing with the notorious inmate.
The ministry admitted liability, and was ordered to pay the legal costs of the case alongside the damages.
Bronson received a two-year sentence in September last year for holding a prison governor in a headlock.
Now known as Charles Salvador, he was first convicted in 1974 and used his time behind bars to establish himself as a marketable artist as well as publishing a book about his unique fitness regime.
A prison officer left with post-traumatic stress disorder after he was attacked by inmate Charles Bronson has been awarded £32,500 in damages.