Carney's game-winner, with 8:39 left in the extra session, was his fourth field goal of the game.
The Bengals' Shayne Graham had tied the contest with a 21-yard field goal on the final play of the fourth quarter.
The Giants, defending Super Bowl champions, are 3-0 for the first time since 2000.
Manning connected on 26 of 43 passes for 289 yards and a touchdown. The scoring pass, a 4-yarder, came with just under 2 minutes remaining in regulation time.
Carson Palmer was 27 of 39 passing for 286 yards and a touchdown for Cincinnati (0-3).
The Bengals' T.J. Houshmandzadeh caught 12 passes for 146 yards and a score.
Notable deaths of 2014 [PHOTOS]
Julius Randle breaks leg during NBA debut