Newton's scores were his 11th, 12th and 13th of the season, bypassing Steve Grogan's NFL mark of 12 rushing touchdowns in the 1976 season for New England while playing under a 14-game schedule.
The Carolina duel threat finished with 204 yards and one interception on 12-of-21 passing in addition to his 54 rushing yards.
Jonathan Stewart added 80 yards and a touchdown on the ground for Carolina (4-8), which took its second straight victory.
The staggering Buccaneers (4-8), meanwhile, lost their sixth straight as backup quarterback Josh Johnson was held to 16-of-27 passing for 229 yards, one touchdown and one interception.
He played in place of Josh Freeman, who was injured in last week's loss to Tennessee.