in 1859 a charter was signed establishing the Zoological Society of Philadelphia, this being the first in the USA. The zoo in itself didn't open until 1874 because that pesky Civil War nonsense got in the way.
The Central Park Zoo opened in 1861, so as an actual facility is the first.
It is a matter of interpretation as to which is the older (of course both will claim to be).
From Wikipedia, regarding the Central Park Zoo:
The zoo was not part of the original "Greensward" design for Central Park created by Olmsted and Vaux, but a Central Park menagerie near New York's Arsenal, on the edge of Central Park located at Fifth Avenue facing East 64th Street, spontaneously evolved in 1859 from gifts of exotic pets and other animals informally given to the Park; the original animals on display included a bear and some swans. In 1864, a formal zoo received charter confirmation from New York's assembly, making it the United States's second publicly owned zoo, after the Philadelphia Zoo, which was founded in 1859.
