Using sensors, authorities rescued 20 people from the debris of the building in Mumbai's crowded Thane district, said K P Raghuvanshi, police commissioner.
The old building came crashing down while most residents were asleep, he said.
The cause was not immediately known, but Mumbai has been hit with heavy monsoon rains.
Nine people were hospitalised, but their injuries were not life threatening, Raghuvanshi said.
Building collapses are common in India because builders often use substandard materials and multistory structures are often built without adequate supervision, and the massive demand for housing around India's cities and pervasive corruption often result in builders adding unauthorized floors or putting up illegal buildings.
Ten days ago, another building collapse in Mumbai left 10 people dead. In April, 74 people were killed when an eight-story building being constructed illegally in Thane caved in. It was the country's worst building collapse in decades.