The fire swept through a building at 41 Spring St., where the suspect, who lives on the second floor, allegedly fought with police and others who arrived on the scene, NBC-TV, New York, reported.
The 45-year-old suspect was identified as Wei Chu Wu, the New York Post reported.
After the fire was set, Wu allegedly told another person, "We're all gonna die," the Post report said.
After exiting the burning building Wu brawled with police and a tourist as rescuers attempted to enter, police said.
"The building is essentially destroyed," FDNY Chief of Fire Operations James Esposito said.
The person who died was on an upper floor and the body was burned beyond recognition, he said.
The injured included seven firefighters, The New York Times reported.
Witnesses reported seeing the suspect start the fire in the hallway of the second floor at about 6:40 p.m., reportedly after arguing with the mother of his child.
Esposito said the fire quickly spread to the roof and engulfed the entire back of the building by the time first responders, which included about 200 firefighters, arrived.
The suspect's child and the child's mother were at the 5th Precinct, NBC-TV said.