Technically speaking, The Bronx-Manhattan High Bridge is the oldest in the City, dating back to 1848. It was built 35 years before the Brooklyn Bridge, but portions of it have been replaced whereas the Brooklyn Bridge is mostly original.
In this great slideshow compilation video from Vimeo uploader Lyndon Lorenz, see shots of the High Bridge from the same angle over many different decades.
The High Bridge was built in 1848 as an aqueduct, and is the oldest bridge in New York City. Closed in the 1970s, the structure today enjoys a renaissance as a linear park, connecting the Bronx and Harlem. In this clip, contemporary video is overlaid with vintage photography courtesy of the public domain collections of the New York Public Library, which are online at digitalcollections.nypl.org. The sectional diagram of the bridge is courtesy of the Library of Congress at loc.gov.