The conductor will take up the appointment in September 2017, following in the footsteps of principal conductors including André Previn, Michael Tilson Thomas, Sir Colin Davis and the current incumbent, Valery Gergiev.
Sir Simon said it was "a very obvious step - and the idea of coming home and having a position in this fantastic city is also a thrill for me".
He is currently director of the Berlin Philharmonic Orchestra, where he has been since 2002 and has three years left on his contract.
He will take up the position at the LSO from September 2017.
Sir Simon, 60, made headlines last month when he told the BBC London's concert halls were not up to international standards.
Speaking ahead of a sold-out London residency with the Berlin Philharmonic at the Barbican, the revered conductor said he would consider a London move if there was a "proper" venue.
"You have no idea how great the London Symphony Orchestra can sound in a great concert hall," he told BBC arts editor Will Gompertz.
However Sir Simon said on Tuesday his move to London was not contingent on the city getting a new hall.
He told reporters he had met the orchestra ahead of the announcement at the LSO's Barbican home: "I said to the orchestra this is my last job.
"At the moment this is a normal five year term with a rolling possibility. We will look on each other as a family and will see where it goes.
"It is on all our minds that it is a long term thing."
He later tweeted via the LSO: "I can't imagine a more inspiring way to spend my next years, and feel immensely fortunate to have the LSO as my musical family."
The orchestra added: "We are thrilled that Sir Simon has accepted our invitation to lead the LSO into the future and welcome him as our music director."
The conductor also outlined his vision for his role, calling for "universal access to music, with children and young people at it's heart".
The LSO is regarded as one of the UK's flagship orchestras and its principal conductor is one of the most high-profile roles in classical music. Others to have held the title previously include Andre Previn, Michael Tilson Thomas and Sir Colin Davis. Its current principal conductor, Valery Gergiev, is to stand down at the end of 2015.
Sir Simon announced in January 2013 he would leave the Berlin Philharmonic Orchestra after his contract expires in 2018. At the time he admitted it had not been an "easy decision".
The conductor said once he takes up his role at the LSO, his family would probably not relocate to London.
"We imagine still living in Berlin. I will be with orchestra for four months in the year and I will have a home in London," he said.
He has a number of concerts with the LSO - which he first directed in 1977 - lined up for July 2015 and 2016.
Sir Simon was born in Liverpool and studied at the Royal Academy of Music in London. He made his Proms debut in 1976 and his opera debut in 1977 at the Glyndebourne Festival.
Between 1980 and 1998, he made his name at the helm of the City of Birmingham Symphony Orchestra. He was knighted in 1994.
Sir Simon appeared at the London 2012 Olympic opening ceremony, when he and the LSO were upstaged by Mr Bean while playing the theme to Chariots of Fire.
(courtesy of BBC)