Sangakkara told the national selectors that he would quit as one-day and Twenty20 captain with immediate effect, but continue as Test captain on an interim basis if required.
"After careful consideration I have concluded that it is in the best long-term interests of the team that I step down now as national captain so that a new leader can be properly groomed for the 2015 World Cup," Sangakkara, 33, wrote to Sri Lanka Cricket.
"This was a decision I made prior to the World Cup. I will be 37 by the next World Cup and I cannot therefore be sure of my place in the team.
"It is better that Sri Lanka is led now by a player who will be at the peak of their career during that tournament."
Sangakkara, a fluent left-hand batsman and wicket-keeper, said he had no plans yet to retire as a player.
"I met with the selectors on Monday and I explained my reasons for my decision and I assured them of my fullest support in helping our new captain settle into the leadership," he wrote.
"In this regard, I have offered to continue as the Test captain for the forthcoming series with England, and possibly Australia, if the selectors believe this would help the new captain and aid the transition.
"I have no plans of retirement from international cricket at this stage and, subject to form and fitness, I would like to be considered for selection in all three formats of the game for the foreseeable future."
Sangakkara has scored 8,244 runs in 94 Tests at an average of 57.25 with 24 centuries. He also has 9,164 runs in 291 one-dayers with 11 hundreds.
Sangakkara said he had enjoyed his two-year stint as captain since taking over from Mahela Jayawardene in March, 2009.
"It has been a true honour and a great privilege to serve my country as captain during the past two years," he said.
"Although bitterly disappointed that we could not win the final, I am very proud of the performances of the team.
"I would like to take this opportunity to thank my team-mates and the coaching staff for their hard work and support."
There was no immediate comment from Sri Lanka Cricket or the national selectors on Sangakkara's decision.