Brees had thrown a touchdown pass in 54 consecutive games, the longest such run of success in the history of the NFL, but the New Orleans quarterback was intercepted five times in one of the most dismal efforts of his career.
Ryan, meanwhile, did not throw an interception and his second-quarter touchdown pass helped the Falcons race to a 17-0 lead.
Atlanta, owner of the best record in the NFL at 11-1, moved to the verge of clinching the NFC South title. One more win for the Falcons or one loss by Tampa Bay, which plays Sunday in Denver, will make Atlanta the division champ for the second time in three years.
A loss by Seattle to Chicago on Sunday would clinch a playoff berth for Atlanta.
The Falcons owned a 17-7 lead at halftime and New Orleans missed a chance to draw closer just before the break when the Saints had a touchdown pass wiped out because of offensive pass interference.
Two field goals by Garrett Hartley in the third quarter brought the Saints within four points and by the end of the period the Saints had outgained the Falcons by more than 100 yards.
Matt Bryant, however, kicked a pair of field goals in the fourth period to extend the Atlanta lead.
Ryan threw a 17-yard scoring pass to Tony Gonzalez in the second period to give Atlanta a 14-0 advantage. He completed 18-of-33 for 165 yards.
Brees went 28-of-50 for 341 yards, but his interceptions continually wiped out New Orleans scoring hopes.
Michael Turner scored Atlanta's first touchdown on a 3-yard run and he finished with 83 yards on 12 carries.
The win avenged the Falcons' lone loss of the season, a 31-17 setback in New Orleans on Nov. 11.