Brees completed 30-of-39 passes without an interception in tying his own NFL record with a ninth straight consecutive game of at least 300 yards. He also equaled his own league mark with a 10th straight contest of completing 25 or more passes.
Darren Sproles ran for a score and reeled in seven passes for 114 yards with another TD while Jimmy Graham added a pair touchdown grabs en route to 100 yards for the Saints, who moved to 4-0 for the first time since their Super Bowl championship year of 2009.
Marques Colston contributed seven catches for 96 yards in the victory.
Ryan Tannehill was intercepted three times and lost a fumble for the Dolphins (3-1), whose three-game winning streak starting the season was snapped.
The Miami signal-caller finished 22-of-35 through the air for 249 yards with a touchdown.
Lamar Miller picked up 62 yards with a score on 11 carries in the loss.