Trailing 25-24 entering the final quarter, White's 11-yard scoring grab from Matt Ryan put the Falcons (5-2) ahead for good. Michael Turner added an insurance score on a 3-yard rushing TD with 4:33 left for Atlanta.
Ryan went 24-of-33 passing for 299 yards and three touchdowns, with one interception. White ended the day with 11 receptions for 201 yards and two scores.
Turner gained 121 yards on 23 carries for the victors.
The Bengals (2-4) were led by Carson Palmer, who was 36-for-50 passing for 412 yards and three touchdowns.
Cincinnati stumbled to its third straight loss despite scoring 29 points in the second half.