The farmer shot one dead and wounded another on Wednesday.
Parks authorities armed with rifles then used a helicopter to track down the injured animal and the third lion.
"We had to obviously kill the wounded lion because it becomes very dangerous," South African national parks spokeswoman Janine Raftopoulos told local media.
"With the last one remaining, because he had come into contact with the farmer's livestock, we assessed the situation and the decision was taken to put (him) down."
She said that if the lion had been returned to the park, it would have tried to escape again and would be a threat to humans and livestock.
Kruger Park, which borders Zimbabwe and Mozambique, is home to about 1,500 lions, and nearly the size of Belgium.
Animals sometimes slip past the barrier fences, especially during the dry winter season.
Two months ago, five other lions escaped from the park. Four were re-captured in neighbouring farms and one is still on the loose.
Officials said animals usually escaped through dry river beds or holes along the fence.