The play that set up the winning kick was a 31-yard pass from Eli Manning to Amani Toomer down the left sideline on a third-and-10 from the Bengals 38.
It was difficult to tell whether Toomer got both feet in bounds.
The Giants hustled to the line of scrimmage and handed the ball to Derrick Ward for a 3-yard run to the Bengals 4, precluding a video review.
The NFL official in the press box conceivably could have seen that the play was OK, and opted to let it stand.
Carney kicked four field goals for New York. Brandon Jacobs scored on 1-yard run and Manning, who was 26-of-43 for 289 yards, threw a 4-yard touchdown pass to Kevin Boss to seemingly give New York a 23-20 victory with 1:50 to play.
However, the Bengals (0-3) drove 71 yards and got a 21-yard field goal from Shayne Graham on the final play of regulation.
There were three scores in the final 4:39. A 17-yard pass from Carson Palmer to T.J. Housmandzadeh gave Cincinnati a 20-16 lead.
Then the Giants went 68 yards in nine plays to retake the lead.
But Palmer drove the Bengals again, hitting Houshmandzadeh on passes of 20, 8, 16 and 9 yards to set up Graham's tying field goal.
Graham also kicked field goals of 22 and 30 yards for the Bengals, who are 0-3 for the first time since 2003, coach Marvin Lewis' first season. Chris Perry, who rushed for 74 yards on 20 carries, also scored on a 25-yard run.
Palmer finished 27-of-39 for 286, with Houshmandzadeh catching 12 passes for 146.
Until the overtime, the Bengals appeared ready to borrow a page from the Giants' road to the Super Bowl last season. After an 0-2 start, the Giants went on the road and beat Washington to turn around their season. New York ended that game with a goal-line stand.
The Bengals, who scored a total of 17 points in losing to Baltimore and Tennessee, took a 13-10 halftime lead by scoring on their final three possessions of the half.
Graham sandwiched field goals of 22 and 30 yards around a 25-yard touchdown run by Perry. The key to the success was Palmer's ability to hit quick swing passes, crossing patterns and flanker screens to Antonio Chatman and Houshmandzadeh against an all-out pass rush that still sacked him four times in the first half - and six times overall.
New York, which squandered early field position, answered Cincinnati's first two scored with Jacobs' short TD and Carney's 24-yard field goal.
Carney tied the game at 13 in the third quarter with a 46-yard field goal, and he put New York ahead 16-13 with a 26-yarder.
