Authorities say Peterson then went to Falconer's home, at 130-71 224 Street in the Springfield Gardens section of Queens, Sunday night.
He was reportedly looking for 16-year-old Tony Jr., but 11-year-old Tyshawn Falconer opened the door instead.
Police say Peterson asked for the older brother and then shot the sixth-grader in the chest at point-blank range.
The boy survived because the bullet missed his major organs, according to doctors.
He was able to provide enough of a description of the gunman for police to produce a sketch, and Peterson was taken into custody around 1 a.m. Friday.
"The shooting last Sunday night was senseless, and could well have resulted in the death of the young victim," Queens District Attorney Richard Brown said. "The shooter was apparently looking for the brother of the victim who was shot simply because he answered the door."
Peterson is charged with attempted murder, assault and criminal possession of a weapon. He faces up to 25 years in prison if convicted.
Peterson has prior arrests for assault and second-degree robbery.
