Redman, playing for the injured Matt Ryan (toe), completed 23-of-41 passes for 243 yards with two touchdowns and no interceptions.
The second scoring pass to Roddy White capped a rally back from a 17-10 second-quarter deficit.
Atlanta's Jerious Norwood scored a touchdown, Jason Elam kicked two field goals, but Michael Turner was held to 33 yards on 12 carries for the Falcons, who had lost four of five.
Tampa Bay's Josh Freeman passed for 250 yards with two touchdowns.
Antonio Bryant caught three passes for 91 yards with one touchdown.
Cadillac Williams scored once, Kellen Winslow Jr. made seven receptions for 81 yards, while Connor Barth kicked a 39-yard, second-quarter field goal for the Buccaneers, who have lost three in a row.