Masterminded by Long & Kentish Architects, the project will see the existing museum in Albert Street, Camden, extended to include the collections currently housed at the Sternberg Centre in Finchley.
The new museum - which will be three times larger than the existing facilities - will boast a ground floor café, a children's area and a 100-seat auditorium. There will also be three additional permanent galleries - the Religion, History and Holocaust Education exhibitions - on the first floor.
Above them, new space will be dedicated to special exhibitions and a new education suite.
Carole Souter, director of the HLF, said: 'We're delighted this substantial grant will now enable the museum to create a fascinating and integrated venue in which to celebrate and learn about Jewish history, culture and religion.
'We were particularly impressed with the museum's commitment to attracting a diverse range of visitors from both London and much further afield.
She added: 'The capital has long been in need of a first-class museum which celebrates the Jewish community and we're confident that the Jewish Museum, with this support, will be able to provide just such a place.'
The grant is understood to be the largest single award ever donated to a Jewish communal organisation, however, the museum still needs another £4.4 million to get the project off the ground.