But while Brees avoided game-turning mishaps with an offense he's led since 2006, Brady suffered some costly cross-ups with his new team.
"I made some just bad, terrible turnovers," Brady said. "I obviously have got to do a lot better job."
The first of Brady's interceptions led to Kamara's 6-yard touchdown run. The second pick thrown by the new, 43-year-old Bucs QB -- who left the New England Patriots in free agency after 20 years and six Super Bowl triumphs -- was returned 36 yards for a touchdown by Janoris Jenkins.
"Win the turnover battle like we did, have a turnover for a touchdown, you're a 90-something percent winner in those games," Saints coach Sean Payton said. "Just a lot of things happened in that game that despite how you play, those things can help you win games. The energy and effort I was pleased with."
The 41-year-old Brees passed for 160 yards and two TDs. He did not turn the ball over and took only one sack. His first TD pass was a dump-off to Kamara from 12 yards out. In the fourth quarter he hit free-agent signing Emmanuel Sanders for a 5-yard score.
"I'd probably say I played awful," Brees said. "Certainly, we are used to playing at a much higher level offensively and especially in the passing game. So, we'll get that back on track."
Still, Brees found satisfaction in beating Brady's Bucs.
"Of course, it's significant and it's unique and I'd be lying to you if I said to you it didn't mean a little bit more," Brees said. "I've got so much love and respect for (Brady) and what he's able to accomplish.
"We played against each other in college, for goodness sake, in 1999," Brees continued. "At the end of the day, I think we would both pinch ourselves if you told us back then we'd have the opportunity to play this long and be a part of so many great teams and so many great moments, historic moments."
Few witnessed this latest bit of history in person, with government leaders and the Saints having agreed that no fans should attend the opener amid the coronavirus pandemic.
That turned the normally deafening 73,000-seat Superdome into more of an echo chamber. When Wil Lutz's field goal put New Orleans up 17-7, the squeaking of pulleys could be heard across the stadium as the net behind the goal post was lowered.
"It sort of felt like we were at a Tampa Bay game," Saints defensive end Cameron Jordan said.
Brady compared the environment to "a scrimmage."
"But obviously, it counts," he added. "We've got to look each other in the eye and all work harder, put more urgency on the things we have to do."
Tight end Jared Cook caught five passes for 80 yards for New Orleans, highlighted by a 46-yard reception down the left side that set up Sanders' TD.
Brady was 15 of 28 passing for 157 yards and one TD through the first 54 minutes. He padded his numbers in garbage time, finishing with 239 yards and two TDs, the second scoring strike going to Mike Evans with 2:41 left.
Tampa Bay's opening drive was promising for Brady, who the Bucs hope can lead them to their first playoff appearance since the 2007 season.
He lofted a 26-yard completion to Chris Godwin along the right sideline, drew a pass interference penalty with a 22-yard attempt to Mike Evans and capped the drive with a 2-yard keeper, after which he emphatically spiked the ball.
It was a fleeting, triumphant moment for the Bucs in a game defined more, in the end, by their new star's mistakes.
Both teams largely stood along the sidelines during the national anthem.
One exception was Saints safety and Players Coalition co-founder Malcolm Jenkins, who trotted to the Saints' bench area after the anthem had finished.
The Saints saw two players activated from the practice squad come up with big special teams plays. Defensive end Margus Hunt blocked a 54-yard field goal attempt. Receiver Bennie Fowler recovered a muffed kickoff at the Tampa Bay 18 in the fourth quarter, setting up a 21-yard field goal that made it 34-17 with 8:01 left.
The Saints were flagged four times for defensive pass interference for 101 yards. All of those calls occurred during the Bucs' first three scoring drives. A 33-yard interference penalty against Tampa Bay at its own 6-yard line set up Kamara's second touchdown of the game. Officials called defensive pass interference six times for 149 yards on both teams.
Buccaneers: Evans played after missing practice the past week with a hamstring injury. His only catch was his late, 2-yard TD.
Saints: Two-time All-Pro WR Michael Thomas appeared to be limping in the fourth quarter of a game in which he had just three catches for 17 yards. New RB Ty Montgomery appeared shaken up after special teams play but was able to walk to the sideline.
Buccaneers: Continue NFC South play when Brady makes his home debut against Carolina on Sunday.
Saints: Take part in the first NFL game played in Las Vegas, the Raiders' home opener on Monday night, Sept. 21.
More AP NFL: https://apnews.com/NFL and https://twitter.com/AP-NFL
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