"Four people are critically injured, fighting for their lives, and there are other serious injuries as well".
Fire officials said they responded to the blaze at the building near the Bronx Zoo in Van Nest at about 5:30 local time (10:30 GMT) on Tuesday. A firefighter was listed among the 19 people suffering minor injuries.
The store is on the ground floor with apartments above.
Another resident escaped with her two children and their dog.
The cause of the fire was not immediately clear.
"Children starting fires is not rare", FDNY Commissioner Daniel Nigro said on Friday, adding that the stairway "acted like a chimney".
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The fire appears to have started in a first floor furniture store.
More than 200 firefighters battled the blaze in bitter cold temperatures, and icicles lined the power lines between the four-story building and the hosepipes used to douse the flames.
It took the fire crews eight hours to control the blaze on the Bronx's Commonwealth Avenue.
"Praying for a swift recovery for all those injured", said Mayor Bill de Blasio on Twitter, thanking the fire department for its response.
Investigators say the fire was sparked by a child playing with a stove.