Kumble, 41, who took 619 Test and 337 One-Day International wickets, succeeds former West Indies captain Clive Lloyd in the role.
In its meeting which concluded here today, the ICC board also agreed to extend the terms of both Mark Taylor (representing former players) and David Kendix (statisticans) for a further term.
Ian Bishop (representing former players) indicated that he was not seeking an extension to his term and the ICC Board unanimously supported the nomination of former England captain Andrew Strauss as his replacement.
“I would like to thank Clive Lloyd for his chairmanship of the ICC Cricket Committee and to his tireless contribution to the game,” ICC president Alan Isaac said in a statement.
“The ICC Board were united in their admiration for his work as chairman. We also thank Ian Bishop for his contribution during his term as well.
“In Anil Kumble, we have a new chairman who has unquestioned experience not only as a player with India but also as an administrator with Karnataka State Cricket Association as well,” said Isaac.
“I am sure that he will carry on Clive’s good work and bring, like Andrew Strauss, contemporary thinking to the committee and both understand clearly the issues facing the modern game.”