The Saints were far from their best in the road game but improved to 9-2 on the season with their third straight win while the Falcons slumped to 2-9 following a fifth consecutive loss.
After leading by a single point at halftime, a Garrett Hartley field goal supplied the only score of a grinding second half for the visitors.
Falcons wide receiver Darius Johnson fumbled early in the fourth quarter while in a scoring position, coughing up a key turnover in the home side's quest for an upset victory.
New Orleans remains top of the NFC South division with the Panthers (7-3) in the midst of a six-game winning run and a game in hand against Miami on Sunday.
Saints quarterback Drew Brees completed 23-of-33 passes for 278 yards and two touchdowns while tight-end James Graham had a 100-yard receiving game from just five receptions.
"We knew what type of game it was going to be. I guess nine of our last 12 games with these guys have gone down to the last possession and it did again today," Brees told reporters.
"They are a great football team despite what their record might say and we knew that.
"Obviously, any time you go on the road in your division it is hard to get a win so we battled on both sides of the ball, played very well, gritted it out and got the win."
Atlanta quarterback Matt Ryan completed 30-of-39 passes for 297 yards but failed to throw for a score.
The home side showed plenty of intent from the outset, marching down the field in the opening drive with Steven Jackson capping it off with a one-yard touchdown run and an early 7-0 Falcons lead.
However, the Saints matched them by scoring on their opening drive, Benjamin Watson the beneficiary of a one-yard pass from Brees to tie it up at the end of the first quarter.
Atlanta kicker Matt Bryant gave his side a 10-7 lead with a 39-yard field goal before Brees found James Graham with a long bomb, and he took it to the end zone for a 44-yard score.
Another Bryant field goal as the half came to a close gave the visiting Saints a slender 14-13 halftime lead.
Hartley extended the advantage to 17-13 with a field goal early in the second half and the four-point margin held firm.
(Reporting by Ben Everill in Los Angeles; Editing by John O'Brien)
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