The home side edged ahead for the first time later in the quarter when Dalton handed the ball off to Giovani Bernard, who sprinted 89 yards, dodging several tackles on the way to the endzone.
The scores were level again early in the third when Fozzy Whittaker powered his way over for his first touchdown since joining the team in the off-season after Antoine Cason had picked off a Dalton pass moments earlier.
The Panthers quickly found themselves back in the red zone, and Newton kept the ball himself to score his first rushing touchdown of the season.
Nugent kicked a field goal late on to put the Bengals back in front, but Graham Gano added a 44-yard effort with the final play of regulation time to send the game into overtime.
The Bengals got the ball first, but could only add another Nugent field goal, giving the Panthers a chance to win the game with a touchdown, but they also had the settle for three points.
Dalton completed a string of passes to Sanu to get the ball into field goal range again inside the final minute, bringing Nugent on for a 36-yard attempt, but he sliced the kick wide of the right upright to leave the scores level as time expired.
The result leaves the Bengals at the top of the AFC North division with a 3-1-1 record, while the Panthers lead the NFC South.