Designed by little-known Belgian architect Robbrecht en Daem, the £12.36 million scheme will almost double gallery space and triple capacity for visiting schools and community groups.
The expansion will include moving galleries into the neighbouring Passmore Edwards library building - which will itself move to a new home in Mile End.
There are also plans to build a street-level restaurant, expand the bookshop and provide public access to the gallery's archive for the first time in its 100-year history.
Whitechapel director Iwona Blazwick said: 'This major commitment from the HLF is fantastic news not just for the Whitechapel's future but for the whole community of the East End. The gallery acts as a beacon for the area bringing the very best art and culture to east London. We can now become an even better resource for the community and attract new visitors.'
The new gallery is expected to open its doors to visitors by 2008.