Plans for the revamp include the introduction of new access paths and the removal of disused or overgrown surrounding roads and paths to create new wildlife habitats.
The park's lake will also be improved and expanded under the scheme. The new plans are in addition to an existing £1.2m investment in Chumleigh Gardens in the centre of the park.
Funding has been provided through a £2m grant awarded to Burgess Park in 2009 through London mayor Boris Johnson's Premier Park competition. Aylesbury New Deal for Communities - now the Creation Trust - provided a further £4m in match-funding.
Creation Trust director Charlotte Benstead said: "These plans will make Walworth a destination and a reason to visit rather than just a cut-through for the residents of Southwark."
Benstead explained that improving the park was a major aspect of attracting investment to regenerate the nearby Aylesbury Estate. "We hope that this adds impetus to getting the area action plan fully implemented," she added.
The mayor said the planning consent represented an important step forward. "Now the work can really begin," he added. "The park will be transformed with better layout, colourful gardens and wildlife areas for local people and visitors alike."
Work is expected to begin in April or May with the first phase of improvements scheduled to be completed by March next year.