Supermarket worker David Sinclair, 34, shook his five-month-old son Joshua in a fit of anger, causing him to die at Hairmyres Hospital hours later.
At the High Court in Glasgow, Sinclair pleaded guilty to culpable homicide in regard to the infant's death on 6 December, 2012.
He is due to return to the court for sentencing next month.
Sinclair was originally charged with murder, but admitted the reduced charge of culpable homicide, just days before he was due to go on trial.
Prosecutor Ashley Edwards said that on the day the child died, Sinclair and his now wife Kirsty and Joshua went swimming and then shopping in East Kilbride.
At 17:30 Kirsty Sinclair went out to meet her mother and sister.
Just 11 minutes later she received a phone call from Sinclair, who sounded "shocked". He told her: "Joshua isn't breathing properly, come home right away." He then hung up and dialled 999.
The court heard that Kirsty Sinclair, who is pregnant, married Sinclair on the anniversary of their son's first birthday on 9 July 2013, and had said she was standing by him.
Initially the cause of Joshua's death was unascertained, but the case was reviewed by a team of medical experts who discovered evidence of trauma in the eyes and brain which could only have been caused non-accidentally close to the time of death.
One of the experts, Dr Peter Richards, consultant paediatric neuro-surgeon at John Radcliffe Hospital, Oxford, said: "This would be compatible with him having suffered a shaking injury involving handling at greater force than encountered in everyday life.
"It would be compatible with a shaking event where a carer, who has not planned to injure a child, has a momentary loss of control with tragic consequences."
Defending QC, Ian Duguid, told the court that Sinclair had found it "unimaginable" that the handling of his child would have resulted in such catastrophic injuries.
The  barrister added: "He is extremely remorseful for the events which have ensued. Witnesses in double figures would have come forward to talk of his caring abilities as a parent."
Det Ch Insp Laura McLuckie, who was the senior investigating officer in the case, said: "David Sinclair admitted to being responsible for the death of his baby son, Joshua, who died needlessly at a young and vulnerable age.
"My sincere condolences go out to Joshua's extended family, who have endured this harrowing incident and continue to live with the pain of Joshua's untimely death.
"A defenceless baby died at the hands of his father, the one person whose role was to protect and care for him."
A man has admitted killing his baby son in a "harrowing incident" at his home in East Kilbride.