It was carrying a small camera, a water bottle and a flare. A "small quantity" of radiation was measured on the drone and is being investigated by police, said Kyodo news.
Abe was in Indonesia attending an Asia-Africa summit when the incident happened on 22 April. An official at the prime minister's office declined to comment.
It was not immediately clear who sent the drone or why. But a Japanese court rejected a bid by residents to stall the restart of a nuclear power plant in south-western Japan on 22 April, dismissing concerns about the safety of nuclear power in the wake of the 2011 Fukushima radiation disaster.
Televised aerial footage from the scene showed the drone with propellers covered under cardboard and later a blue tarpaulin. Dozens of police officers were seen carrying it away.
Japanese broadcaster NHK said an official at the premier's office found the drone and that the device was around 50cm in diameter.