Danny Platt confessed, told police where to find the child's body and has been booked on a charge of first-degree murder in the death of Ja' Shawn Powell, police said.
"He had said he would kill either his wife or his child before he paid child support," which he recently had been ordered to do, Riley said.
Riley said he did not know the amount of child support and would not describe how the boy was killed, saying the coroner would do that after the autopsy was complete. The coroner's spokesman did not immediately return a call.
"The mother is in a safe place," Riley said.
Although he had visiting rights, Platt, 22, of New Orleans, had never visited the boy until he picked him up Friday, Riley said.
Police put out a notice Saturday asking people to look for the boy and saying his father had told them three men with dreadlocks and AK-47 rifles had piled out of an SUV and kidnapped Ja' Shawn shortly before midnight Friday.
"His story never really added up," Riley said. "He was a suspect from the very beginning."
Riley said Platt eventually confessed and told officers where to find the body.
Police spokesman officer Janssen Valencia said he did not know if Platt has an attorney.
Platt had only a couple of "very minor" previous arrests, Riley said.
"How does an individual - because he's ordered to pay child support to take care of a kid ... believe that this is so much pressure that he would face - he would do this hideous act to his own child, or to any child, and think that is a remedy to paying child support?" Riley said. "I mean there are some sick individuals in this society, and this gentleman is clearly one."