They say that a church custodian found the crying infant wrapped in towels in the indoor nativity scene.
The laws of New York allow for unwanted babies to be dropped off anonymously at churches, hospitals, police or fire stations without fear of arrest.
The newborn has been taken to hospital where he is in good health, police say.
Police say they are searching for whoever left the child because the terms of the Abandoned Infant Protection Act stipulate that a child who is being dropped off must be be left with an appropriate person.
The child was abandoned while a church custodian was out for lunch.
He came back to find the crying child wrapped in towels in the manger of the nativity scene.
Police have been questioning people in the area and are reviewing surveillance camera footage.
"The beautiful thing is that this woman found in this church, which is supposed to be a home for those in need, this home for her child," Father Christopher Ryan Heanue, one of the priests at the church, told the Reuters news agency.
"A young couple in our parish would love to adopt this child and keep this gift in our community. It would make a great Christmas miracle."
A newborn baby with his umbilical cord still attached has been found lying in a manger at New York's Holy Child of Jesus church, police say.