The poet, who was defamed by Soviet authorities and ultimately emigrated to the West, spent his teens and 20s in the flat shared by several families in the centre of Saint Petersburg, Russia's second city, which was then called Leningrad.
"Joseph has left a moving essay called 'A room-and-a-half' about this flat where he lived with his parents between 1955 and 1972," said Nina Popova, one of the museum's organisers, who also runs a museum dedicated to poet Anna Akhmatova.
The new museum, which will open to the public on Sunday, occupies most of the rooms in the former communal flat, a form of shared housing that was ubiquitous in the city's centre in the Soviet era.
Four families shared a single kitchen and bathroom in the apartment. The Brodsky family occupied two rooms, though most of one was converted into a darkroom by the poet's father.
Brodsky lived there until he left Russia for the United States in 1972 under pressure from the KGB secret police, after a series of defamation campaigns, stints in a psychiatric hospital and months of exile in the north of Russia.
"We tried to create a space that has the ambience of that era, of Joseph's life with his parents as he was becoming a poet," Popova said.
Original furnishings include Brodsky's typewriter and even a table scratched by his cat.
Some of the rooms have an eclectic otherworldly feel. The former darkroom has bare cement walls and monochrome photographs floating in a bathtub with eerie red illumination.
Creating the museum was a lengthy process after the idea was first proposed in 1998 by arts figures including the director of the Hermitage Museum Mikhail Piotrovsky and legendary cellist Mstislav Rostropovich.
That was partly because the museum's creators had to persuade the people still living in the communal flat to move out.
"We had to find new housing for the inhabitants of this apartment. That took a long time and required a lot of money" which was finally provided by private sponsors, Popova said.
One resident refused to move and her room had to be partitioned off from the rest of the apartment and given a separate entrance.