The Kenya Red Cross said a search team managed to pull three children and an adult from the remains of the six-story building in the Huruma area.
The Red Cross said 150 building units and adjacent homes were affected.
Live TV footage showed the National Youth Service and firefighters removing stones by hand and a crowd cheering as a child was removed from the rubble.
Hours-long traffic jams caused by flooded roads delayed rescue teams trying to reach the scene, said Japheth Koome, Nairobi's police chief, who confirmed the death toll. Kenya is in the midst of its April-May rainy season.
Jacob Kiruma, who said he lived in the house adjacent to the one that collapsed, said the building was constructed "shoddily."
The structure had been built in less than five months and the 126 single rooms were quickly occupied at a rent of $35 a month, Kiruma said.
Area legislator Stephen Kariuki said this was the second building to collapse in a year. He blamed the county government of failing to follow through with demolitions of buildings that were identified as unfit for humans.
Because of a high demand for housing in Nairobi, some property developers bypass building regulations to cut costs and maximize profits. The Architectural Society of Kenya has estimated that 50 percent of structures in Nairobi are not up to code.
President Uhuru Kenyatta last year ordered an audit of all the buildings in the country to see if they are up to code after eight buildings collapsed, killing at least 15 people.
The report from the audit by the National Construction Authority found that 58 percent of buildings in the capital were unfit to live in. The majority of Nairobi's people live in low-income areas or slums.
See more photos after the cut...
Correspondent: Oscar
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