The Falcons (11-1) will clinch the NFC South with a month to go if Tampa Bay loses at Denver on Sunday. They also dealt a big blow to the Saints (5-7) and their fading playoff hopes.
William Moore had two of the five interceptions, which were the most of Brees' career and came four days after he had two passes picked off and returned for touchdowns in a loss to San Francisco.
"That's the first time that's ever happened to me, so that's extremely disappointing," Brees said. "I pride myself on being a good decision-maker and not someone who will be a detriment to the game."
Brees had thrown a touchdown pass in 54 consecutive games, breaking Johnny Unitas' long-standing record earlier this season. There was an apparent scoring pass to Darren Sproles late in the first half, but it was nullified by a penalty.
"I didn't realize that until we walked off the field," Falcons coach Mike Smith said. "That's an unbelievable streak. Drew Brees is an outstanding quarterback. The way the defense played tonight speaks volumes. The guys had gone out there and thrown touchdown after touchdown game after game after game."
After the Sproles TD was wiped off the board, Brees made another huge mistake with New Orleans inside the Atlanta 10, allowing the clock to run out without at least attempting a field goal.
When the Saints arrived in Atlanta, their bus was pelted by eggs at the airport, epitomizing the long rivalry between the teams. New Orleans had dominated in recent years, winning four in a row and 11 of 13.
This time, Michael Turner scored on Atlanta's opening possession, Tony Gonzalez hauled in a touchdown pass from Matt Ryan, and Matt Bryant booted three field goals, including a 55-yarder.
The defense did the rest. Thomas DeCoud, Sean Weatherspoon and Jonathan Babineaux also had interceptions for Atlanta, and another pick by Corey Peters didn't count because of a penalty.
Brees had a couple of games with four picks, but nothing like this. He finished 28 of 50 for 341 yards.
The defending NFC South champion Saints lost their second in a row and will likely have to win out to have even a faint hope of making the postseason during a tumultuous year that was marred by a bounty scandal and a season-long suspension for coach Sean Payton.
After winning so many close games, the Falcons started this one as if they were intent on a rout.
Ryan completed a pass on the first play from scrimmage, then turned it over to a running game that has struggled most of the season. Turner burst around right end for a 35-yard gain. Jacquizz Rodgers broke off two straight 14-yard gains. Finally, it was Turner going in standing from 3 yards out, giving Atlanta a quick 7-0 lead.