The quango's decision represents a major u-turn - it had previously been committed to demolishing the 1964 building, designed by London County Council architects.
The LDA has long wanted to knock down the Modernist building as part of a 'regeneration' initiative - a move that was attacked by many in the conservation lobby, including English Heritage.
But in a change of heart, while still insisting that the building's setting must be transformed, it is no longer seeking permission to demolish.
Previously, the organisation had even gone as far as to put the demolition call to a local vote, an initiative which proved that the public backed sending in the wrecking crew.
Julian Harrap, who has worked on a scheme to retain the building, described the LDA's decision as 'splendid news'.
'We are really very pleased, it's a wonderful building that needs to be kept,' he said.
'While they haven't taken everything that we wanted to do in the surrounding park into account, the fact that they are retaining the sports centre means the most important thing has been achieved.'
And the Twentieth Century Society's Cordula Zeidler echoed these views. 'We are excited that we have won our campaign to keep it,' she said. 'We really are delighted.'