A jury at Newcastle Crown Court was told that Michael Davison died in hospital last October four days after being injured when his father was looking after him while the boy's mother, Diane Smedley, 17, was staying with her mother.
Davison, of Washington, Tyne and Wear, originally denied a charge of murder but when his trial began on Tuesday he changed his plea to guilty to manslaughter.
The court was told that six weeks before his death, Michael was taken to Queen Elizabeth Hospital, Gateshead, with head bruising and seen by Dr Marietta Higgs, one of the paediatricians at the centre of the 1987 Cleveland child sexual abuse controversy.
After X-rays failed to show any fracture, Dr Higgs decided the injuries were accidental and the boy returned home.
However, after Michael's death from brain damage from the shaking, evidence was found of a fracture which may have been caused during the August incident in which the father claimed the baby had banged its head against a door, Paul Worsley QC, for the prosecution, told the court.
The jury was told that this damage to the membrane over the child's brain, which was discovered by a pathologist during the post-mortem examination, was the sort of injury which might not show in an X-ray.
The post-mortem examination also revealed that Michael had several other old injuries which had not previously come to light - healing fractures to five ribs and two leg fractures.
Mr Worsley told the court that after Michael's death, Davison had asked Ms Smedley to try to prevent a post-mortem examination being conducted.
Sentencing Davison, Mr Justice Potter said: 'This is an inexcusable case where death has resulted from the gross mistreatment of a baby.'