The Bucs dominated time of possession thanks to two 75-yard scoring drives plus converting on an incredible 10 of 11 third-down attempts.
Winston had three completions to Mike Evans on the drive, then found Godwin, a Penn State rookie, open in the end zone for his first National Football League touchdown.
Brees completed 5-of-6 passes for 57 yards during the 13-play, 92-yard drive, including a 3-yard touchdown to running back Zach Line with 7:13 remaining.
Rookie running back Alvin Kamara accounted for both New Orleans touchdowns in the first two quarters, when the Saints assumed a 14-13 advantage at halftime.
The touchdowns were the 13th and 14th of the season for Kamara, who has eight rushing and five receiving in addition to the second 100 yard-plus kickoff return for a TD allowed by Tampa Bay in two weeks. Saints' safety Vonn Bell had just forced the Tampa Bay punt when he sacked quarterback Jameis Winston for an 11-yard loss. The New Orleans finished 22 of 30 for 245 yards, one touchdown and no interceptions.
The Saints failed on an attempted fake field goal when kicker Wil Lutz was tackled one yard short of the first down. It drove 92 yards in 13 plays and took up almost seven minutes for a touchdown and 24-20 lead with 7:07 to go in the fourth quarter at Raymond James Stadium.
The Buccaneers (4-11) scored on a pair of 1-yard runs, the first by Peyton Barber and the second by quarterback Jameis Winston. Adam Humphries had seven catches for 102 yards.
Buccaneers: S T.J. Ward left with a knee injury and did not play in the second half.
Buccaneers: Coach Dirk Koetter, who's 13-18, was told before the finale that he will return for a third season.