Brees pushed the Saints forward to score with their very first possession on an 82-yard drive, with Darren Sproles finishing in a strong run through the middle.
Miami got on the scoreboard before the first quarter finished as Caleb Sturgis was successful with a 32-yard field goal, but Ryan Tannehill's fumble when play resumed put the visitors back under pressure.
The Saints took full advantage of that mistake to score their second touchdown in some style, with Brees fooling the Dolphins defence with a play-action fake, before finding Jimmy Graham in the endzone with a precise pass.
Tannehill recovered from that mistake to find Marcus Thigpen in a clever play, setting a screen to find the blocker on a 50-yard drive.
Lamar Miller finished things up by driving home from close range, moving the Dolphins to within four points.
Miami looked to seize on that positive drive soon after as they gained possession, but another Tannehill error put the visitors in trouble with an interception.
Once again the Dolphins were made to pay for turning the ball over as Brees found Sproles for the second time, catching Nolan Carroll out of position to make sure of his third score just before the interval.