Brady, who joined the Buccaneers in March after calling time on his 20-year reign as New England Patriots' quarterback, was sacked three times and intercepted twice as the Saints powered to victory.
It was an uncomfortable outing for the 43-year-old Brady, widely regarded as the greatest quarterback in NFL history.
Although Brady got off to a fairytale start -- scoring the game's opening touchdown after a one-yard sneak into the end zone -- the Buccaneers were chasing the game for long periods thereafter.
The Saints drew level through an Alvin Kamara touchdown and then took the lead after Brady's first interception of the day. Running back Kamara grabbed his second touchdown.
A Wil Lutz field goal put the Saints 17-7 ahead at half-time, and New Orleans stretched their lead to 24-7 early in the third when Brady was picked off by Janoris Jenkins, who returned for a touchdown.
Although the Bucs closed the gap to seven points with a field goal and a touchdown, Saints quarterback Drew Brees found Emmanuel Sanders for another touchdown to leave the home team in control.
- Double delight for Newton -
Elsewhere Sunday, Cam Newton rushed for two touchdowns as New England made a winning start in their post-Brady era with a 21-11 win over the Miami Dolphins.
Newton, who joined the Patriots in the off-season from the Carolina Panthers, extended his quarterback touchdown rushing record to 60 with a score in each half at Foxborough, Massachusetts.
Former NFL Most Valuable Player Newton's first touchdown came in the second quarter, skipping into the end zone from four yards out. He added a second touchdown in the third quarter to extend New England's lead to 14-3.
The Dolphins closed the gap to 14-11 with a Jordan Howard touchdown followed by a successful two-point conversion attempt early in the fourth quarter.
But Newton then led the Patriots on a 10-play, 75-yard drive that ended with running back Sony Michel barrelling over from one yard out to make it 21-11.
Newton, 31, is aiming to relaunch his career under Patriots coach Bill Belichick after an injury-plagued final two seasons with the Panthers, who jettisoned the quarterback in March.
- Ravens rout -
Reigning NFL MVP Lamar Jackson threw for 275 yards and three touchdowns as the Baltimore Ravens demolished the Cleveland Browns 38-6.
Jackson opened his account for the season with a five-yard touchdown pass to Mark Andrews as the Ravens took the lead in the first quarter.
Although the Browns responded with a touchdown through David Njoku from Baker Mayfield's pass, the Ravens rattled off four more touchdowns without reply to pull clear of their AFC North divisional rivals.
In the late game, Malcolm Brown rushed for two touchdowns and Jared Goff passed for 275 yards as the Los Angeles Rams opened their new $5 billion SoFi Stadium with a 20-17 win over the Dallas Cowboys.
The Green Bay Packers made a winning start in their NFC North divisional battle with the Minnesota Vikings, romping to a 43-34 victory in Minneapolis.
Packers quarterback Aaron Rodgers led the Green Bay offense, passing for 364 yards and four touchdowns with no interceptions. Rodgers made 32 completions from 44 passing attempts.
In Atlanta, the quarterback duel between Seattle Seahawks star Russell Wilson and Falcons counterpart Matt Ryan finished with Wilson firmly in the ascendancy as the Seahawks scored an impressive 38-25 win.
Ryan had a whopping 450 passing yards and two touchdowns, but Wilson threw for four touchdowns and 322 yards in the win.
In Detroit, Lions linebacker Jamie Collins found himself ejected from his team's 27-23 defeat to the Chicago Bears in bizarre circumstances after he appeared to lower his helmet and make contact with an official.
Collins looked to be attempting to complain about Bears running back David Montgomery lowering his helmet into a tackle, but inadvertently brushed the game official's chest with his head as he did so.
Collins' misery was compounded by a fourth quarter collapse that saw the Bears reel off 21 unanswered points to turn a 23-6 deficit into victory.