Tight end Josh Hill was in the endzone a few plays later, with Brees spotting him open at the back of the zone.
The Vikings had to change their gameplan in the second quarter as Matt Cassel scrambled while trying to get Minnesota back into contention, injuring his ankle in the process to bring rookie quarterback Teddy Bridgewater onto the field.
Bridgewater led a couple of promising drives, resulting in two field goals for Blair Walsh to keep Minnesota in contention at the break.
The kicker was on target again early in the third to cut the Saints' lead to just four points, while an injury to New Orleans' starting center left Brees struggling with the snap.
The quarterback recovered in the fourth quarter to fire a long pass downfield for Colston, the 64th time that the quarterback has completed a touchdown pass to the wide receiver to end any hopes of a Vikings comeback.
The result gives the Saints their first win of the season.