They extended this advantage in the second quarter as Dalton threw a shuffle pass to Giovani Bernard, who proceeded to weave his way through a number of tackles before reaching the end zone.
Dan Carpenter struck a field goal with 4:19 left in the first half to make it a seven-point game at the halfway stage, but Dalton's third touchdown pass of the day, this time to Marvin Jones, put the Bengals in control in the third quarter.
The Bills hauled themselves back into the contest with 10 minutes left of the fourth quarter when Lewis looped a pass over to Scott Chandler, who had a clear run into the end zone.
Lewis continued to impress in the closing stages as he threw an inch-perfect 40-yard pass to Marquise Goodwin, who had got on the wrong side of his man and had an easy finish to force the sides into overtime.
The Bills could not build on their strong fourth quarter during overtime, however, and it was the Bengals who got the vital score as Nugent struck a 43-yard field goal with 6:49 of the additional period remaining to secure the victory for his side.