The Saints (11-4) already have a playoff spot secured but can win the NFC South Division with a victory over Tampa or if Carolina loses to Atlanta.
The Bucs would strike again a couple drives later as Winston got the touchdown on a keeper and great second effort to break the plane. The Bucs dominated time of possession thanks to two 75-yard scoring drives plus converting on an unbelievable 10 of 11 third-down attempts.
Winston shook off three interceptions against the Saints and ended with 363 yards on 28 of 51 passing.
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.
The Bucs were able to hold the Saints' one-two punch of a ground game to 30 total yards in the first half.
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.
Dirk Koetter will return to coach Tampa Bay Buccaneers next season
The Buccaneers won't be giving the quick hook to a second consecutive head coach after only two seasons. He has thrown 47 touchdown passes against Tampa Bay, his most against any opponent in the NFL.
The Bucs connected on a 42-yard field goal on their next drive to cut New Orleans lead to 24-23 with 4:31 left on the clock.
Tampa Bay had rallied for a fourth-quarter lead, only to see the Saints grab the lead back.
But Saints return man Tommylee Lewis was stripped by Tampa Bay's Josh Robinson and the fumble was recovered by Isaiah Johnson, who scooped it up and scored from the 7. That led to the punt to the 5-yard line, which Winston converted into a massive opportunity.
The Saints got a big game from rookie Alvin Kamara, who returned the Bucs first kickoff 106 yards for a touchdown, tying the game at 7-7.