Newton was 18 of 37 for 195 yards and ran for 77 yards on nine carries.
The fifth-year quarterback provided the highlight of the day when he raced up the middle on a draw play and tumbled over defensive back Rahim Moore - nearly sticking the landing - for a 3-yard TD that put Carolina ahead 17-10 late in the third quarter.
Newton also threw TD passes of 25 yards to Ted Ginn Jr. and 36 yards to Philly Brown for Carolina (2-0).
Ryan Mallett, starting in place of Brian Hoyer, threw for 244 yards and a touchdown and ran for a 6-yard TD for the Texans (0-2). Hoyer was benched after committing two turnovers in a Week 1 loss to Kansas City.
The Texans trailed 10-3 at the half after being held to 93 yards and six first downs against a Carolina defense playing without top defender Luke Kuechly, who sat out with a concussion.
Mallett went to the no-huddle offense and the Texans took the opening second-half kickoff 80 yards, with tight end Garrett Graham making an a one-handed grab in the back of the end zone on a third down pass to help tie the game.
On Carolina's next possession J.J. Watt sacked Newton - his seventh straight game with a sack dating back to last year - and mocked Newton by pointing the opposite direction for a first down.
The crowd booed - and Newton responded.
A few plays later, the quarterback saw Moore in his sights as he approached the line of scrimmage and took to the air, flipping his body. He went over the goal line and nearly landed on his feet, but was hit by Jared Crick and stumbled a bit before beginning his celebration.
A.J. Klein, filling in for Kuechly, intercepted Mallett early in the fourth quarter and Newton cashed in with a prefect strike to Brown, who outleaped Jonathan Joseph for the 24-10 lead. Brown had struggled in the preseason with dropped passes, but has been perfect so far during the regular season.
Mallett would run for a TD on fourth-and-4 cut the lead to 24-17.
The Texans had one last chance to tie the game after Crick Graham Gano's 53-yard goal was blocked, but Mallet's fourth down pass sailed out of the end zone in the closing seconds.
Make sure to catch a complete wrap-up of today's game on Inside the Game at 10:35pm on ABC13.