The fire erupted in a mobile shop on the second floor of the mall, which mainly houses shops for mobile accessories, besides a few other stores.
The BMC said around 3500 residents of the 55-floor Orchid Enclave tower adjoining the mall were shifted as a precautionary measure.
A police officer said they evacuated around 300 people from the mall when the blaze erupted.
The fire was initially categorised as level 1 (minor) but was upgraded to level-3 at 10.45 pmand to level-4 at 2.30 am, after flames spread to other parts of the building.
Mumbai Mayor Kishori Pednekar visited the site and reviewed the fire fighting operation.
According to fire brigade officials, the exact cause of the fire was not yet known.
Earlier on Thursday, a fire broke out in a garment factory at Kurla in Mumbai. It was doused after two hours.