Since then the popular game - involving placing the numbers 1 to 9 in each row, column and square of a 9 by 9 grid - has spread around the globe.
Tournaments take place across the world and it is estimated that millions play versions of the game each day.
On Monday Nikoli published a statement on its website saying Kaji had died at his home in Tokyo from bile duct cancer on 10 August 2021.
Kaji was born in the northern Japanese city of Sapporo in 1951.
After dropping out of Keio University, he set up the puzzle magazine Nikoli, which published its first edition in August 1980.
The origins of Sudoku are unclear. Some credit the 18th Century Swiss mathematician Euler as its creator, while others say it came to the Arab world from China via India in the 8th or 9th Century, according to the New York Times.