Turner racked up 107 yards on 24 carries and scored twice in the first half as the Falcons (6-2) built a 17-7 lead.
But the Bucs made a strong bid to tie the game late in the fourth quarter. Trailing by six points, Josh Freeman (189 yards, two touchdowns, two interceptions) guided Tampa Bay 89 yards on a 12-play drive to the Atlanta 2-yard line with 2:37 remaining.
But the Falcons the stopped LeGarrette Blount on a 4th-and-1 and hung on for the win.
Matt Ryan completed 24-of-36 passes for 235 yards and a touchdown in the Falcons' victory.
Freeman was picked off twice for the Buccaneers (5-3), who dropped their fourth straight game to Atlanta.