The deceased and his accomplices were fleeing from Kruger National Park (KNP) rangers on Saturday when they encountered the elephants.
Realising they had been spotted by park officials, the group attempted to escape before unexpectedly coming across the herd.
One of the suspected poachers sustained an eye injury but managed to escape, KNP said in a statement.
A second was arrested while the deceased was discovered "badly trampled" and died as a result of their injuries.
The group left behind an axe and a bag of provisions when they realised they were being chased.
The arrested suspect was apprehended with assistance from the park's Airwing and KNP units. A rifle was also discovered and the case has been referred to the police.
The search for the suspect who managed to escape is ongoing.
KNP's managing executive, Gareth Coleman, said he was proud of his team's work and that it is unfortunate that a life "was unnecessarily" lost.
"Only through discipline, teamwork and tenacity will we be able to help stem the tide of rhino poaching in KNP," he added.
Mr Coleman called on the local community to come forward with any information that might help authorities help find the suspected who escaped.
In a separate incident at the same park in April 2019, a poacher was trampled to death and then eaten by a pride of lions.
A human skull and a pair of trousers were found after a search party was launched to find the poacher, whose accomplices said was trampled by an elephant.
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