The Buccaneers remained winless in seven games this season, joining Jacksonville as the only teams in the NFL without a victory.
Carolina improved to 4-3 with its third straight triumph and is 1 1/2 games behind New Orleans for the lead in the NFC South.
Newton put the Panthers in front for good with a 1-yard throw to Greg Olsen at the end of a 70-yard march that took 8:52 off the clock in the opening period.
DeAngelo Williams then gave Carolina a 14-6 halftime lead by scoring on a 12-yard run. The Panthers drove 80 yards in six plays to the Williams touchdown.
The Panthers dominated the second half with Newton running 6 yards for a touchdown in the third period and throwing 3 yards to Mike Tolbert for another score in the fourth.
Newton completed 23-of-32 passes for 221 yards without an interception.
Tampa Bay's only score came with 2:53 to play when Mike Glennon hit Timothy Wright with a 10-yard pass.
Glennon, making his fourth consecutive start, went 30-of-51 for 275 yards.