Mogale and Beukes said the victim's family had not yet identified the body.
Pistorius made history in London last year when he became the first double-amputee track athlete to compete in the Olympic Games, propelling him to the status of an athletics superstar.
Having had both his legs amputated below the knee before his first birthday because of a congenital condition, he campaigned for years to be allowed to compete against able-bodied athletes. Having initially been banned because of his carbon fiber blades — which critics said gave him an unfair advantage — he was cleared by sport's highest court in 2008 and allowed to run at the top events.
He competed in the 400 meters and on South Africa's 4x400 relay team at the London Games, making history after having his selection confirmed on South Africa's team at the very last minute. He also retained his Paralympic title in the 400 meters in London.
South Africa's Sports Confederation and Olympic committee released a statement on Thursday saying they had been "inundated" with requests for comment but were not in a position to give out any details of the shooting.
"SASCOC, like the rest of the public, knows no more than what is in the public domain, which is there has been an alleged fatal shooting on the basis of a mistaken identity and an apparent assumption of a burglary," the South African Olympic committee said. "The organization is in no position to comment on the incident other than to say our deepest sympathy and condolences have been expressed to the families of all concerned."
The International Paralympic Committee also said it wouldn't comment in detail apart from offering its condolences to the victim's family.
"This is a police matter, with a formal investigation currently underway," the IPC said. "Therefore it would be inappropriate for the IPC to comment on this incident until the official police process has concluded. The IPC would like to offer its deepest sympathy and condolences to all families involved in this case."
South Africa has some of the world's highest murder rates, with nearly 50 people killed each day in the nation of 50 million. It also has high rates of rape, other assaults, robbery and carjackings.
U.N. statistics show South Africa has the second highest rate of shooting deaths in the world, second only to Colombia.