Scheduled to open in 2016, the £1.8 million gallery will accommodate shows and residencies from British and international artists, as well as creating exhibition opportunities for students at Goldsmiths, which is part of the University of London.
Assemble saw off five other shortlisted studios, including South London Gallery designer 6a Architects and Jerwood Gallery architect HAT Projects, with a proposal designed to "expose the hidden character" of Victorian bathhouse Laurie Grove in Lewisham.
The Grade-II listed building, which has housed studios and classrooms since being taken over by Goldsmiths in 1999, will now be extended to accommodate a series of top-lit galleries built around the historic water tanks.
"We envisage the gallery becoming a new centre for the arts in south London," explained Paloma Strelitz and Adam Willis from Assemble, whose past projects include a cinema under a motorway flyover and a scaly studio building.
"The Victorian bathhouse at Laurie Grove offers a series of extraordinary found spaces," they said.
"The cast iron water tanks have a powerful materiality which will be preserved and amplified, whilst new top-lit galleries will provide a rich spatial counter-point in an ensemble offering unique opportunities for the display of art."
Assemble's proposal was selected by a panel of judges that included architect David Chipperfield and artist Antony Gormley, who described the project as "an important moment in the development of Goldsmiths".
"This will become a resource for the university and for London: a place where students and the wider public can experience and test-drive new forms of art, as well as see relevant examples of art from the past, ancient and modern," said Gormley.
"It will be a place where curators can exercise their skills, artists both international and from the college can make, exhibit and discuss their work and where all the ways in which Goldsmiths extends our understanding of the culture of our time can be shared with a curious public."
The gallery will be entirely paid for by donations, with the majority of funds expected to be raised from an auction at Christie's in February 2015. The college will offer a collection of artworks from Goldsmiths' alumni, such as Bridget Riley, Damien Hirst, Antony Gormley, Sarah Lucas and Gary Hume.