The Saints started the ball on offense and moved downfield early on but were kept to a field goal attempt that was missed by Garrett Hartley.
Rex Ryan's side used their good field position to good effect but could only put three points on the board following a defensive stand by the Saints near their end zone, and Nick Folk made no mistake blasting his effort through the posts from 39 yards.
Brees immediately responded for the Saints and handed them the lead as he threw deep down the right-hand sideline for Jimmy Graham, who beat Jaiquan Jarrett in coverage and managed to shrug him off to break the goal-line for the 51-yard score.
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The Jets drove downfield but were again held to a field goal from Folk and Brees punished them with a scoring drive of his down and he found Graham in the end zone for the second time of the match on an out route for his 10th touchdown of the season.
However, Ryan's side showed resilience as they pulled out a scoring drive to fightback in the game which resulted in Chris Ivory running in for a three-yard score to bring his team within a point.
Things got better for the home side when Brees was intercepted for the second time in the contest. Antonio Cromartie made an athletic catch to pick him off down the sideline, which allowed Geno Smith to rally his side to take the lead at the half, as he ran for the touchdown himself from three yards out.
After the break the Jets went further ahead through the boot of Folk, who added two more field goal from 47 and 45 yards in between a 55-yard effort from Hartley to give the home side a nine-point lead early in the fourth quarter.
Late in the game Brees drove his team downfield but he was forced to settle for three from the boot of Hartley to cut Ryan's side's advantage to six with two minutes left.
The Saints quarterback tried to get the score his side needed to get ahead in the game but the Jets brought pressure to get to him to force four incompletions to win the game.