"The way we've been able to (get turnovers) and score is a big reason we're where we're at right now," Payton said.
Brees hit Marques Colston for an 18-yard score. Thomas scored on a 22-yard run in the first quarter and a 1-yard catch out of the backfield with 3:03 to go, flipping backward over a tackler and into the end zone. That touchdown made it 35-24 and led to Superdome chants of "Who dat say they gonna' beat them Saints?"
"Pierre did a good job of getting through that last defender," Payton said. "It was following the series where he fumbled and it was good to see him respond."
Reggie Bush had a 1-yard TD run late in the first half, giving the Saints a lead they would not relinquish.
Roddy White beat Greer on a 68-yard scoring pass from Ryan early in the third quarter, and the Falcons pulled to 28-24 on Elam's 25-yard field goal with 11:33 to go.
That field goal, however, came only after Payton sprinted down the sideline and launched his red flag about 20 yards, just in time to challenge a tying touchdown catch by White in the back of the end zone. Replays showed White allowed the ball to touch the turf as he bobbled the catch.
Atlanta's second consecutive loss dropped the Falcons (4-3) three games behind the Saints in the NFC South. It marked the first time the Falcons had lost two straight under second-year coach Smith.
"It's a disappointing loss but it's a long season," Ryan said. "There's a lot that can happen between now and Week 17. We've just got to take care of our business."
Atlanta also squandered Michael Turner's best game of the season. He had 151 yards rushing, including a 13-yard touchdown.
The Falcons' defense also produced a score. Thomas DeCoud's jarring sack on a delayed blitz up the middle dislodged the ball from Brees and Kroy Biermann returned it 4 yards, giving Atlanta a 14-7 lead late in the first quarter.
Brees also was intercepted on a spectacular leaping grab by Brent Grimes. The QB never got rattled though, completing 15 of his first 17 passes and leading three scoring drives in the first half.