The victims included the acting head of the Kazakh federal border service Turganbek Stambekov and his wife, the statement said.
The plane crashed about 20 kilometres (12 miles) from Shymkent airport where it had been due to land after a flight from the capital Astana.
Kazakhstan's KTK television said the plane fell from a height of 800 metres (2,600 feet).
Footage of the crash site aired on Kazakh state television on Wednesday showed only fragments of the An-72 military transport plane remaining on the ground.
Khabar state television also cited local witnesses as saying that a heavy winter storm had descended the region at the time of the crash.