NEW ORLEANS — The Atlanta Falcons can forget about perfection, thanks to their resurgent rivals from the City that Care Forgot.

Of all the teams to end Atlanta’s unbeaten run, it had to be the New Orleans Saints. And it had to be a memorable, drama-filled game that left the Falcons frustrated by the 1 meager yard they could not get to take a lead inside the final 2 minutes of a 31-27 loss Sunday.

“It’s very frustrating,” said Falcons coach Mike Smith, who is 2-7 against the Saints and 49-15 against the rest of the NFL in his four-plus regular seasons as Atlanta coach. “We were close. ... This will be something we will learn from.”

If the Falcons (8-1) want to beat the Saints (4-5) in their rematch in Atlanta on Nov. 29, it would help to learn how to stop dynamic young tight end Jimmy Graham, who caught seven passes for a career-best 146 yards and two touchdowns.

Still, Atlanta might have pulled it out, but Jabari Greer made a diving, touchdown-saving pass breakup on a late fourth-down pass intended for former James Island High star Roddy White.

“It’s not like they came out here and won a game today,” White said. “I think we kind of gave it to them. ... We play them in three weeks and we’ll be ready.”

Greer’s play represented a measure of redemption for the veteran, who’d given up two long passes earlier in the game that led to Atlanta scores.

“I was kind of depressed before that play,” Greer said. “I saw my name on the waiver wire. ... Being able to go out there and make a play for our team and be able to celebrate after everything that happened and just to win that way, it’s a gift.”

Matt Ryan was 34 of 52 for a career-best 411 yards and three touchdowns with one interception. The game was there for him to win when his 9-yard pass to Harry Douglas gave Atlanta a second-and-goal at the 1. But Ryan could not connect on a second-down pass for Gonzalez, the Saints stuffed Michael Turner for a 1-yard loss on third down, and then came Greer’s play.

“I could put that ball in a better spot — a little higher and more in front of him to give him a chance to catch it,” Ryan said. “We’ve always played them tough and haven’t had the outcomes we wanted.”

Atlanta’s Tony Gonzalez finished with 11 catches for 122 yards and two scores for Atlanta, becoming the first tight end to catch 100 touchdown passes.