The soldiers were advancing towards rebel fighters outside the capital, Bangui, but some rebels managed to cut around them and attack their base, said Major Jiyane, whose first name could not be immediately obtained.
He told a memorial service for the fallen soldiers in Pretoria: "We stood our ground and fought, the grass and the bushes of Bangui provided cover for us in the dark hours of the night."
He said while the soldiers were fighting on the ground, a chopper had provided aerial support for them.
He described the soldiers as "disciplined tigers".
"The soldiers waited like tigers for my command and fought like lions until the end."
The 13 soldiers were killed in a clash with rebel fighters outside Bangui last month.
Twenty seven were wounded and have since been flown back to South Africa for medical treatment.
The wounded soldiers were admitted to 1 Military Hospital in Pretoria.
A representative for the soldiers' families said on Tuesday the SANDF had provided a chaplain and social workers to help them deal with their grief.
"We were angry when we received the news about the deaths of our brothers, but the support from the SANDF made us stronger."
He appealed to the media and public to show respect for the soldiers' families, and to allow the soldiers to be buried in a dignified manner.
The SANDF last week said the families would decide whether they wanted a military funeral, a semi-military funeral, or a private funeral.