Yes, in nearly every case. Real Property Law 235-b, the warranty of habitability, is part of every residential lease in New York whether it is written in or not, and it requires the landlord to keep the apartment and the shared common areas fit for human habitation and free of any condition that is dangerous, hazardous, or detrimental to the tenant's life, health, or safety. A roach or mouse problem is that condition.
I have been treating apartment buildings on Long Island for thirty five plus years, and the first thing I tell a new property manager is that the argument about whose fault it is usually costs more than the treatment. The statute is short. The landlord warrants the premises are habitable. The tenant cannot sign that away, because the law says any agreement to waive it is void as against public policy. If a tenant withholds rent over roaches and you end up in front of a judge, the judge is reading 235-b.
Long Island buildings have a second layer. The Multiple Residence Law covers cities under 325,000 people and every town and village in the state, which is all of Nassau and Suffolk, and it applies to any dwelling occupied by three or more families living independently. Section 174 of that law says the owner must keep the whole building and the lot in good repair and clean, and specifically free of pests and rodents, along with dirt, garbage, and anything else that could endanger life or health. That is not a tenant-complaint standard. It is a standing duty on the owner.
New York City has its own version in Multiple Dwelling Law 80, which requires the owner to keep the building, the yards, courts, and passages "clean and free from vermin," and in the Housing Maintenance Code, which I get to below. The short version for a manager with buildings in Massapequa and Jackson Heights: the duty is yours in both places, and the City adds paperwork.
apartment and property manager pest control