The department said there were 22 non-life-threatening injuries reported to civilians and one injury to a firefighter.
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. The other victims suffered minor injuries.
Another resident escaped with her two children and their dog.
The store is on the ground floor with apartments above.
"Numerous people were brought out of the building by the firefighters on scene", said New York City fire chief Daniel Nigro.More news: Mariah Carey Dick Clark Productions Says ... Give Her The Chair!!
The cause of the fire was not immediately clear. On the frigid morning, firefighters were battling not only the flames but the cold as well.
Five hours later, the fire crews were still fighting to control the blaze on the Bronx's Commonwealth Avenue.
The child's mother said in the aftermath of last week's fire that her son had a history of playing with the stove in his family's first-floor apartment, Reuters reported.
That blaze started after a three-year-old boy was playing with a stove. New York City Mayor Bill de Blasio called it the worst fire tragedy the city has seen in a quarter century.