Witnesses said one mother threw her children out of windows to save them after the fire began late on Wednesday.
Officials believe about 22 people from an extended family of immigrants from Mali lived in the brick building.
Up to 19 people were injured in what is said to be one of the city's worst fires in recent times.
Some of the injured are in a critical condition, and there are fears that the death toll could rise further.
'Great tragedy'
The fire started shortly after 2300 local time (0400 GMT Thursday), witnesses and officials said.
"It just seems more painful and unfair when children die - everyone around them dies a little bit, too"
They said the flames spread quickly throughout the overcrowded home making escape for so many of the residents impossible.
Neighbour Edward Soto said he had seen a woman throwing a child out the window and then jumping herself - the only way they could escape the inferno.
"When she threw the first kid out, he fell right on me and I fell over. When she jumped, it seemed like she broke her leg as soon as she hit the floor. There was blood everywhere," Mr Soto was quoted as saying in the New York Post.
Firefighters arrived on the scene within minutes after receiving the call and extinguished the fire shortly after 0100 on Thursday (0600 GMT), officials said.
The residents of the building in the Highbridge area of the south Bronx are all believed to be members of one extended family of immigrants from Mali.
The children ranged in age from less than a year old to 10. They included seven-month-old twins and also five children from another family.
Investigators say an electric heater in the basement may have been the cause of the fire.
New York Mayor Michael Bloomberg said the loss of young lives was a great tragedy.
"It just seems more painful and unfair when children die. Everyone around them dies a little bit, too," he said.
The mayor said that - excluding the 11 September 2001 attacks - this was the single worst loss of life in a New York City fire since 1990.
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