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The Saints had the better of a tight opening quarter, taking a 6-0 lead courtesy of two Shayne Graham field goals from 30 yards and 29 yards respectively.
Tampa Bay set about reducing the 13-point deficit in the closing stages of the first half, with Patrick Murray the first to get on the scoreboard for his side with a 55-yard field goal.
The visitors then cut the gap to just three points with less than 30 seconds remaining of the half as Louis Murphy just about got his feet down in time having been found by a 20-yard Mike Glennon pass.
The comeback was completed in the early stages of the third quarter as Tampa Bay moved into the lead courtesy of a nine-yard Bobby Rainey run straight through the middle.
Things got even better for the Buccaneers shortly afterwards as the ball slipped out of Brees's hand while he was going for a pass, allowing Danny Lansanah to intercept it and return it 33 yards for a touchdown.
The Saints responded with just under five minutes left in the third quarter when Brees picked out Travaris Cadet with a five-yard pass to reduce the gap to four points.
However, the Buccaneers struck once more in the early stages of the fourth quarter, Glennon and Robert Herron combining for a nine-yard score.
The Saints hit back again shortly afterwards as Thomas somehow found a way through for a 27-yard rushing touchdown, but Brees's pass to Benjamin Watson for the two-point conversion was unsuccessful.
It became a three-point game with less than seven minutes remaining, however, as Glennon was sacked in his own endzone by Junior Galette for a safety.
Graham added a 44-yard field goal shortly afterwards to level the scores at 31-31, and that was how it stayed until the end of regulation time.
It was the hosts who struck first in overtime as Khiry Robinson raced into the endzone for an 18-yard score to clinch an important win for his side and maintain their perfect home record this season.