Sonn, aged 57, had spent several days in a Cape Town hospital following complications after undergoing a minor colon operation.
He was the first African to head up the ICC but his death leaves the organisation without an obvious successor after England's David Morgan and India's Sharad Pawar tied in a recent vote to replace him.
ICC chief executive Malcolm Speed said: "Percy's mantra was that the game should be inclusive not exclusive; he relished modern cricket's diversity"
Sonn's predecessor as ICC President, Ehsan Mani said: "As a cricket administrator and a man, Percy Sonn was a giant.
"In all the circles in which he moved, he commanded a huge amount of respect and that was never more obvious than when he was in an ICC Board meeting.
"Percy never spoke for the sake of it but when he did speak people listened. He was one of the most intelligent men I have ever met and cricket will be much the poorer for his passing," the Pakistani added.
"It is a tragedy that he was only able to fill the role of ICC President for one year."
Cricket South Africa (CSA) President Ray Mali said: "This is a terrible shock and a devastating piece of news as I have lost a close personal friend.
"I know Percy was so proud to represent South Africa and the whole continent of Africa as the ICC's first President from this part of the world and he filled the role with great dignity and strength."
Australia's World Cup-winning captain Ricky Ponting added: "I am shocked and saddened to hear this news and first and foremost my thoughts are with Percy's family and friends.
"I will always associate Percy with one of the happiest moments of my career as he was the man who handed over the Cricket World Cup trophy to the Australia team at the end of the tournament in Barbados last month.
"He and his wife then flew with us back from Barbados to London where we went our separate ways and to think he is no longer with us less than a month later is a huge shock.
"I have been told of his lifetime of service to the game in what, for many years, must have been difficult circumstances in South Africa. Cricket obviously owes him a huge debt of thanks."
Born on 25 September 1949, Percival Henry Frederick Sonn, known as Percy, dedicated much of his life to cricket and started his career as an administrator while still a teenager with his club side, Bellville South.
Sonn became ICC president in June 2006 after successfully heading the United Cricket Board of South Africa where he played an integral part in the integration of South Africa into the world game after the end of apartheid.
Away from cricket, Sonn was a lawyer and became a legal advisor to the South African Police Service.
He leaves his wife Sandra and three children - a daughter and two sons - as well as his mother, six brothers and a sister.
The ICC said details of his funeral would be made public as appropriate. The process for choosing a new president of the ICC is that under the organisation's articles the ICC board will appoint an acting president until an election can be held.
The board is due to meet in London next month. Details of the form any election for a new president will be announced in due course, the ICC said.