That New Orleans connection was just too much for the Chicago Bears.
Brees threw two touchdown passes, Graham tied an NFL record with another 100-yard game and the Saints beat the Bears 26-18 on Sunday to remain unbeaten.
"We knew the formula coming into this game was to remain patient, to run the football effectively, to be very efficient in the passing game and to take care of the football, priority No. 1," Brees said, "and we were able to do all those things."
Brees was 29-of-35 passing for 288 yards in his first victory in four career games at Soldier Field. Garrett Hartley matched a career high with four field goals as New Orleans (5-0) picked up its first win in Chicago since a 31-10 victory on Oct. 8, 2000.
Graham continued his torrid start for the Saints, catching 10 balls for 135 yards in his fourth consecutive 100-yard game -- matching an NFL record for a tight end. Tony Gonzalez was the first to accomplish the streak in 2000, and Graham matched the surge in 2011.
Graham and Brees said they spent more time together during the offseason, and it looks as if all that work is paying off.
"I'm just blessed to be with such a great quarterback and a guy who has so much trust in me," Graham said.
Graham helped the Saints hold onto the ball for 36 minutes, and Brees' team had no turnovers. The ball control was crucial with Alshon Jeffery putting together a career day for the Bears (3-2).
Jeffery had 10 receptions for a franchise-record 218 yards, breaking Harlon Hill's mark of 214 at San Francisco on Oct. 31, 1954. He also had a 3-yard touchdown grab in the second quarter.
"What I did out there, I'm more concerned about the 'W,' not about what I did," he said. "It's a team thing. It took 11 guys to get 200-plus yards or whatever I had."
The Bears got the ball back with 21 seconds left and no timeouts. Cutler then passed to Jeffery in the middle of the field for 21 yards, but time expired before they could run another play.
"It's just a few plays here and there," said Cutler, who was 24 of 33 for 358 yards and two touchdowns in Chicago's second straight loss. "Against a team like that, the way they played offensively and ate up the clock it's hard to rebound if you miss three or four plays in the game."
It took a while for Brees to get started, but New Orleans' defense made sure he had plenty of time to get warmed up.
The Saints sacked Cutler twice while limiting the Bears to one first down in the first quarter. Chicago had just 51 yards when it got the ball back with 5:57 left in the first half.
After two field goals by Hartley -- including a 19-yard chip shot after Cutler fumbled on a sack by Malcolm Jenkins deep in Chicago territory -- Brees started to find his rhythm. He connected with Graham on pass plays of 29 and 38 yards to get New Orleans to the 7 with 7:23 left in the half.
This time, the Saints got into the end zone.
Brees rolled out on second down and flipped the ball to Pierre Thomas, who fought through D.J. Williams' tackle attempt and extended his right arm over the goal line for the 2-yard touchdown that made it 13-0 with 5:57 remaining.
"When my name is called to do the job, I go out there and do my best," said Thomas, a Chicago native. "That's all I can really ask for. When my name is called, just go out there and perform."
The Bears responded with a seven-play, 80-yard drive that ended with Jeffery's TD. But they gave the ball back to Brees with 2:41 left in the half, plenty of time for the star QB to direct another scoring drive.
Thomas had a big 2-yard run on fourth-and-1 from the Chicago 27, and then went 25 yards for the score on a screen play with 23 seconds to go.
Thomas' second TD reception made it 20-7 and was the result of a perfect call by Payton, who still has a deft touch when it comes to play-calling after missing last season as a result of the bounty scandal.
"It was a screen play away from the pressure," Brees said. "So you had everybody going one direction and here we are throwing a screen and getting linemen out. The key is your ability to get linemen out and down the field.
"That was a big play. Big momentum gainer going into halftime."
The Saints also won a road game against the Bears in 2002, but that victory came in Champaign because Soldier Field was being renovated. ... Bears DT Nate Collins hurt his left knee in the third quarter and did not return. ... Bears DT Stephen Paea was inactive due to a toe injury. Landon Cohen got the start just 10 days after he was signed off waivers.