Bayern have acted just hours after their dramatic Bundesliga title triumph on the final day of the season.
Vice-chairman Jan-Christian Dreesen will take over from Kahn, while Salihamidzic's replacement is yet to be appointed.
A statement on the club's official website said: "Oliver Kahn is no longer CEO of FC Bayern Munchen AG. Hasan Salihamidzic will also be replaced as board member for sport. This was decided by the club's supervisory board.
"Kahn will be replaced by the current vice-chairman of the board Jan-Christian Dreesen. Salihamidzic's successor is still to be decided."