Turner, who finished with 121 yards on 26 carries, scored on a 2-yard run with 3:35 remaining, giving Atlanta a 27-17 lead.
Rookie quarterback Matt Ryan completed 16-of-26 passes for 194 yards, two touchdowns and one interception.
Green Bay quarterback Aaron Rodgers, playing despite a gimpy right shoulder, competed 25-of-37 passes for 313 yards, three touchdowns and one interception.
Ryan Grant rushed for 83 yards on 18 carries, and wide receivers Greg Jennings and Donald Driver each scored once in the Packers' third straight loss.
They fell to 2-3 for the season.