"We're rolling and we can't stop," said Cameron Jordan, who led the way as New Orleans sacked Atlanta's Matt Ryan five times. "We've got to get better."
Brees threw a 44-yard touchdown pass to Jimmy Graham and a 1-yarder to Benjamin Waton, giving the first-place Saints (9-2) a sweep of the season series with their NFC South rivals.
Atlanta (2-9) dropped its fifth straight game, clinched its first losing season since 2007 and was officially eliminated from the division race with five weeks remaining.
The Falcons had a chance to pull off the upset, driving to the Saints 29 with less than 3 minutes to go. But New Orleans turned up the pressure on Ryan, Atlanta missed a long field goal, and the Saints ran off all but the last 5 seconds.
New Orleans also forced a crucial fumble early in the fourth quarter. Keyunta Dawson stripped the ball from Falcons rookie receiver Darius Johnson, and Corey White fell on it at the Saints 13.
"We knew coming here was going to be a challenge," Saints coach Sean Payton said. "The two teams, the history, regardless of the record it has always been a close game, and tonight was no exception."
The Georgia Dome rocked with chants of "Who Dat! Who Dat!" as the final seconds ticked off the clock, the Saints ensuring they will remain alone atop the NFC South and just one game behind Seattle for home-field advantage throughout the NFC playoffs. New Orleans will face the Seahawks on Dec. 2 in what shapes up as one of the biggest games of the year.
"The identity of the team is still developing," said Jordan, credited with 2½ sacks.
The Falcons had hoped to salvage a bit of pride in a lost season, driving from their own 9 actually, the 3 after Ryan was sacked to the Saints 29. But Ryan was sacked again by Jordan and hurried into a pair of incompletions. The Falcons surprisingly sent on Matt Bryant to try a 52-yard field goal.
He knocked one through, but the Saints called timeout just before the snap. His next try sailed wide of the left upright.