New Orleans had been 0-5 in postseason games outside of the Big Easy since entering the league in 1967. The Saints (12-5) will play at NFC top-seed Seattle next Saturday; they lost there 34-7 in the regular season.
"It's loud, it's crazy, they've got a good thing going there," Brees said of the next challenge. "Obviously, they've only lost one game there in the last two years. But having been there less than a month ago, I think that serves us well, what to expect, how to prepare for it.
"But we're going to need our best game, that's for sure."
This was not their best offensive game, by far. But Graham, now with his 10th team after being signed by the Saints just over two weeks ago to replace long-time kicker Garrett Hartley, also connected from 36, 46 and 35 yards.
"I didn't feel an ounce of fear," the 13-year veteran said. "If I had been here for 14 years or for one game, my job is the same. I feel like I've been adopted into a family."
Brees threw a 24-yard touchdown pass to Lance Moore, Mark Ingram rushed for 97 yards and another score, and the Saints' defense slowed Chip Kelly's up-tempo offense just enough.
The Eagles wound up 10-7 in Kelly's first year as coach. He guided them from worst to first in the NFC East, but they were only 4-5 at home.
"I didn't think winning the division was a surprise to us and we're disappointed we didn't move forward," Kelly said.
Nick Foles hit rookie Zach Ertz for a 3-yard touchdown with 4:54 remaining as Philadelphia rallied from a 13-point deficit to take a one-point lead. But Darren Sproles had a 39-yard kickoff return and a horse-collar tackle brought New Orleans to the Philadelphia 48. Using mostly runs, the Saints ate up the clock and set up Graham's winner.
"We know what we're all about," Brees said. "This was a great testament to that. Coming on the road, hostile environment, great team, one of the hottest teams in football and getting one of those big victories."
Two teams with prolific offenses couldn't get on track for much of the game.
Graham's 46-yarder as the first half ended made it 7-6 as everyone wondered where all the offensive fireworks went. The teams combined for an average of 816.7 yards and 53.5 points per game during the regular season.
When Riley Cooper, who earlier scored Philadelphia's first touchdown, had a huge drop in the third quarter, the Saints immediately made Philly pay. A 66-yard drive highlighted by Benjamin Watson's 27-yard reception led to Ingram's TD from the 4 for a 20-7 lead.
That's when the Eagles finally made a dynamic play with the ball, DeSean Jackson's leaping 40-yard catch over Corey White. NFL rushing leader LeSean McCoy scored from the 1 on fourth down to make things close.
Jackson helped make it closer with a 29-yard punt return that set up Alex Henery's 31-yard field goal early in the fourth quarter, cutting the Saints' lead to 20-17.
Another 40-yard completion, from Brees to seldom-used Robert Meachem, led to Graham's 35-yard field goal, setting up the frantic finish.
Jackson was the central figure in setting up Ertz's go-ahead score, drawing a 40-yard pass interference call on the beleaguered White.
But Brees then guided the winning drive, disappointing Foles, who went to the same Texas high school as Brees and wears the same No. 9.
"This doesn't feel good," said Foles, who led the NFL in passer rating and was 8-3 as a starter. "We wanted to keep playing. It's tough to lose. It hurts. But we're going to keep building."
Meanwhile, the Saints move on with confidence in a defense that held the NFL's top rushing offense to 80 yards McCoy had just 77 and their own ground game, which gained 185 without injured Pierre Thomas.
"Just the fact they had confidence in me to carry the load, it says a lot," Ingram said. "People are going to talk, you'll always have critics, have doubters. But my teammates know I go out there and bust my tail every day and that's all that matters."
NOTES: Brees finished 20 for 30 for 250 yards and was picked off twice in the first half. ... Foles was 23 of 33 for 195 yards. ... Eagles TE Brent Celek injured his neck, but could have returned had the Eagles gotten the ball back late in the fourth quarter. ... New Orleans cornerback Keenan Lewis suffered a head injury in the third quarter and didn't return.
AP NFL website: www.pro32.ap.org