The bridge here was finished in 1865. There's even a plaque on the bridge with that date. The Brooklyn Bridge was finished 4 years later.
http://www.cincinnati-transit.net/suspension.html
Cincinnati was the first major city in "The West", or what is today known as the Midwest. Its 1850 population of 115,435 ranked sixth in the United States, far ahead of nearby St. Louis (77,860) and Chicago (29,963). Its cross-river neighbors of Covington and Newport were home to 20,000 more residents, and as ferry traffic increased, the necessity of a bridge became apparent. The 1849 charter issued by the state of Ohio required no piers in the river, a length of 1,400ft., and a deck clearance of 112ft. By the time construction started in 1856, a revised charter eased the required length to 1,000ft. and height to 100ft. The ensuing period of construction stretched over a decade, interrupted by financial shortages and the Civil War, during which the city and unfinished bridge were under threat of attack.
The bridge opened to pedestrians in December 1866, and the 1,057ft. main span was at that time the longest in the world, surpassing the Wheeling, WV suspension bridge (1849).
Not only was the Cincinnati Suspension Bridge the world's longest, but it was also the first to utilize both vertical suspenders and diagonal stays fanning from either tower. This advance was next seen on the Brooklyn Bridge (also designed by John Roebling), which surpassed the Cincinnati bridge in length and almost every other statistical category in 1883.