The boy was kept in the hospital for observation for 90 minutes after he was found to have a fever but then was released with prescriptions for antibiotics and pain relievers.
On Monday evening, the boy began vomiting and was taken back to the hospital for evaluation.
Texan congressman Joaquin Castro called for a congressional investigation into the boy's death.
The father and son were not identified, and the agency said it will release more details "as available and appropriate".
Alonzo is the second child to die in CBP custody this month.
She became sick on the bus ride to a border patrol station and arrived with a 41-degree Celsius fever.
Border Patrol said they want to make it easier for medical options, including access to factions like the Department of Defense, the U.S. Coast Guard, FEMA, the CDC and others.
He is locked in a battle with Congress over funding for his planned border wall, which he claims will stem migration from Latin American countries plagued by gang violence and poverty.
"While the Customs and Border Patrol notified Congress within 24 hours as mandated by law, we must ensure that we treat migrants and asylum-seekers with human dignity and provide the necessary medical care to anyone in the custody of the U.S. government", Castro said.
The Guatemalan foreign ministry called for an investigation "in accordance with due process".
An eight-year-old boy from Guatemala died in USA government custody on Tuesday, immigration authorities say.
Late at night, Felipe and his father were transferred to the Border Patrol station in Alamogordo, New Mexico, because of overcrowding at the El Paso station.
It was the second death of a child in the agency's care this month.
Oscar Padilla, the Guatemalan consul in Phoenix, said he was told by the boy's father in a telephone interview that the two had been travelling from their home in Nenton, a village about 450 kilometres from Guatemala City. Agents cleaned up the vomit but say the father refused further medical assistance.
The children's deaths have highlighted the life-and-death challenges that migrants face in attempting to come to the U.S. and caused CBP to announce several policy changes around its care of children under 10 years old.
Apparently the most recent child to die did receive an initial medical screening.
In a new, lengthy statement, the US Customs and Border Protection (CBP) said the boy, whose identity has not been officially confirmed, died at 23:48 local time (06:48 GMT) on 24 December.
The boy's death came on the same day that Jakelin Caal, a Guatemalan migrant girl who died in US custody under similar circumstances earlier this month, was buried. Her body was taken to Guatemala for burial.
Her death marked another flashpoint in the debate over the White House's hard-line approach to immigration enforcement, with many - including Jakelin's family - wondering if better medical care may have saved her.
The Trump administration has tried to deter asylum seekers from crossing the border irregularly between ports of entry, warning that they could face arrest, prosecution and deportation.
"CBP is coordinating with the Centers for Disease Control on the numbers of children in custody as well", he said.