The match was suspended after 26 minutes and the venue was evacuated following the incident. According to the club's report there was no violence or third party intervention.
"A supporter fell from the Sivori Alta stand and died on the spot. The medical service arrived instantly in the area as did the police and various security agencies," the club said in a statement.
"The stand where the deceased had his season ticket was at 90% capacity. At the time of the fall, there was no third party intervention. It was also found that there was no violent situation in the stand or around him."
River Plate later named the supporter as Pablo Marcelo Serrano.
The club said the security agencies and the Fiscal Unit Specialised in Mass Events have begun investigations and have closed the stand for 24 hours to obtain evidence.