The 36-year-old man was charged late on Tuesday at the Royal Adelaide Hospital, before being taken into surgery for a self-inflicted neck wound, Superintendent Graeme Adock told reporters.
The mother rang police about 8.15am saying her husband had gone missing from their Stepney home with their son and she had concerns about their welfare.
About 9am, a police patrol was attending an unrelated crash scene at Athelstone when a man in a black Volkswagen Amarok stopped at the scene and told police his son was injured.
"The police officers went to the vehicle and found a small three-year-old child very badly injured," Superintendent Adock said.
"The ambulances were already at the scene, so the child was immediately attended to by the paramedics."
He was taken to Adelaide Women's and Children's Hospital, where he died.
Police believe the boy received the fatal injuries in the family car.
"He received the injuries while he was in the vehicle," Superintendent Adock said.
The man, who had non-life threatening injuries to his neck, was taken to hospital.
Superintendent Adock said it was too early to say whether mental health issues were involved in the tragedy.
A post-mortem on the boy will be held on Wednesday.
"He had injuries consistent with being stabbed, but whether he was stabbed or not we do not know," he said.
"As you can imagine, the mother of the child is very distressed.
"Police are providing her with all the support we can, so it is difficult to get the information we need from husband and wife for various reasons."
He understood the family was not known to police and was not aware of any intervention orders.
"Everyone who is involved in this, the mother and very close relatives and friends, are extremely distraught."
"I just visited the police officers involved. They are very upset. The ambulance officers, they are very upset."
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