The case was filed after city officials posed as 12-14 year-olds on Facebook and were immediately targeted by paedophiles. They were also able to access pornographic images and videos.
"My office is concerned that Facebook's promise of a safe website is not consistent with its performance in policing its site and responding to complaints," said Cuomo.
"Parents have a right to know what their children will encounter on a website that is aggressively marketed as safe."
Cuomo claimed that Facebook had consistently ignored complaints from investigators posing as the parents of the 'children', and that when the company did respond the time taken was "excessive".
While the investigators found that Facebook usually removed offensive material within a week, it was less active when it came to shutting down user groups which displayed and exchanged pornography.
Cuomo cited an example of an investigator who created a profile for a 14 year-old female high-school student from New York on 30 August.
Approximately one week later, the 'female' received a Facebook message from a 24 year-old man asking 'Do you have any nude pics?'.
The investigator lodged a complaint with Facebook pretending to be the student's mother and complaining that her daughter was being solicited by older men.
Facebook has taken no further action, however, and the 24 year-old man's profile is still available on the Facebook site, according to Cuomo.
The court action will force the company to hand over all complaints made against users, the company's response and any public statements it has made about the safety of the site.
Darktrace pushes machine learning to take some of the pressure off of IT and security teams
Google also gets its hands on HTC's IP in a non-exclusive deal
Microsoft, Google and Samsung all targeted as Avast admits to the scale of the CCleaner compromise
Not all loose ends tied yet, admits Bain backer SK Hynix