Sangakkara, 41, will be the first non-British president of the MCC and his one-year stewardship will begin on 1 October, 2019.
"It is a huge honour," said Sangakkara.
"2020 is going to be yet another significant one in cricket, especially at Lord's, and I am thrilled I am going to be able to play a part in supporting its future as president of MCC."
The MCC, who are based at Lord's, remain the guardians of the laws of the game and uphold the spirit of cricket.
Sangakkara's tenure will include two Tests at Lord's - against West Indies and Pakistan - and the launch of The Hundred competition.
During his playing career, Sangakkara scored 12,400 Test runs, more than 14,000 one-day international runs and almost 1,400 T20 runs for Sri Lanka.
He succeeds Anthony Wreford in the role.