The six-time Super Bowl champion, who joined Tampa Bay in March after ending a 20-year career with the New England Patriots, threw for 217 yards with one touchdown from 23 completions.
"It feels good to win," Brady said afterwards. "It's going to feel good to watch the film and learn.
"There was a lot of good things and definitely things we can be better at. So it was fun to win and we've worked hard at winning. We're going to take it and we're going to try to be better next week."
The Buccaneers had taken control of the contest in the first half, racing into a 14-0 first quarter lead with a close-range rushing touchdown from Ronald Jones before Brady connected with Mike Evans from 23 yards for a second score.
Newly signed running back Leonard Fournette then barreled over from one yard out to make it 21-0 at halftime.
Panthers running back Christian McCaffrey grabbed two touchdowns to close the gap to 21-14 but a late Fournette touchdown, rushing 46 yards to score made the game safe for Tampa Bay.
Elsewhere on Sunday, the Super Bowl champion Kansas City Chiefs were pushed all the way before snatching a 23-20 overtime victory against the Chargers at SoFi Stadium in Los Angeles.
The 22-year-old dazzled on his debut, scoring a touchdown from four yards out after leading a 79-yard opening drive, and finishing with 311 passing yards.
Herbert's heroics were not enough to hold off the Chiefs however, who eventually sealed victory in overtime with Harrison Butker's 58-yard field goal.
Mahomes was the savior for the Chiefs, engineering the team into field goal position at the end of regulation, allowing Butker to tie the game at 20-20, before setting up the win in overtime.
- 'Legatron' saves Dallas -
Elsewhere on Sunday, the Dallas Cowboys produced a thrilling fourth quarter fightback before kicker Greg Zuerlein nailed a 46-yard field goal with only four seconds remaining to secure a 40-39 victory over the Atlanta Falcons.
Dallas coach Mike McCarthy, who had been in the firing line after the Cowboys' opening week loss to the Los Angeles Rams, looked to be in for more hard questions as the Falcons opened up a 39-24 lead in the fourth quarter.
Atlanta quarterback Matt Ryan had been in blistering form, tossing four touchdowns and no interceptions for 273 yards.
But Dallas, who also trailed 20-0 in the first quarter, saved their best for last, with a jaw-dropping one-handed catch by Amari Cooper from quarterback Dak Prescott's pass sparking the fightback.
Prescott hit Dalton Schultz for a 10-yard touchdown with just under five minutes remaining before rushing from one yard out to cut the Falcons' lead to 39-37.
Dallas then successfully recovered an onside kick to regain possession from the restart, and Prescott drove his team into position for Zuerlein, the reliable kicker nicknamed "Legatron."
"We dug ourselves such a big hole in the first quarter and I just think the way the guys rallied and just kept going ... I'm very proud of everybody," McCarthy said. "I can't say enough about the way our players kept playing."
Elsewhere on Sunday, veteran quarterback Ben Roethlisberger threw two touchdowns and 311 passing yards in the Pittsburgh Steelers 26-21 defeat of the Denver Broncos.
There was controversy in the New York Jets' 31-13 home defeat to the San Francisco 49ers at MetLife Stadium in East Rutherford in New Jersey.
Five 49ers players left the game with injuries, including defensive star Nick Bosa, with what was feared to be a season-ending knee injury.
San Francisco coach Kyle Shanahan later revealed players had complained about the "sticky" playing surface at the arena.
"That's as many knee injuries and ankle stuff and people getting caught on the turf as I've ever been a part of," Shanahan said. "It was something our players were concerned about right away."