The Buccaneers missed the chance to wrap up a play-off spot.
Victory would have clinched the reigning Super Bowl champions a play-off place but three Brett Maher field goals were enough for the Saints to beat their NFC South rivals.
The last time a Brady-led team failed to score a point in a game he started was back in 2006.
The seven-time Super Bowl winner turned the ball over twice and was sacked four times, while Tampa’s star running back Leonard Fournette and key wide receivers Chris Godwin and Mike Evans all suffered injuries during the game.
New Orleans’ victory meant the Dallas Cowboys have to wait another week to book their post-season place despite a comfortable 21-6 win over NFC East rivals the New York Giants earlier on Sunday.
The Green Bay Packers became the first team to secure a place in this season’s play-offs after beating the Baltimore Ravens 31-30.
Baltimore fought back from 31-17 down in the fourth quarter but an unsuccessful two-point conversion in the final seconds allowed Green Bay to clinch a third straight NFC North title.
Packers quarterback Aaron Rodgers threw three touchdowns to take his career tally to 442 and tie hall-of-famer Brett Favre’s franchise record.
Another team to miss out on booking their place on Sunday was the Arizona Cardinals, who were stunned 30-12 by a Detroit Lions side that went into the game with just one win all season.
In each of the NFL’s two conferences, the four division winners are joined in the post-season by the three teams with the next best records.
And the San Francisco kept their grip on one of those spots in the NFC by beating the Atlanta Falcons 31-13.
In the other conference, the AFC, the Cincinnati Bengals moved top of the North division after beating the Denver Broncos 15-10.
But the Pittsburgh Steelers’ 19-13 win over the Tennessee Titans – and the Ravens’ defeat – means all four teams in the AFC North are within a win of each other, with the Cleveland Browns still to play this week.
The Miami Dolphins are just one win away from the wild card places after they beat the New York Jets 31-24 – their sixth straight victory that took them to 7-7.
Miami’s AFC East rivals Buffalo Bills hold the third and final wild card slot on 8-6 after Josh Allen threw three touchdown passes in a 31-14 win over the Carolina Panthers.
Elders, the Jacksonville Jaguars now hold the worst record in the league, putting them in position to own the number one draft pick for the second successive season, after their first game since Urban Meyer’s sacking as head coach ended in a 30-16 defeat to fellow strugglers the Houston Texans.