Pete Carroll's side have now claimed seven victories on the spin, taking their overall record in this season's NFL to 11-1-0.
The Seahawks were in the endzone again soon after, as Russell Wilson found Miller for a three-yard score.
Six minutes into the second quarter, Brees went some way to making up for his earlier error as he fired a strike to Jimmy Graham for a two-yard touchdown.
However, the hosts restored their comfortable advantage as Hauschka nailed another field goal, before Baldwin added a third touchdown with 13 seconds of the first half remaining.
The scoring was concluded midway through the third quarter when Wilson's attempted pass to Kellen Davis was broken, but fell into the hands of Coleman, who was right next to the tight end and ran in for an eight-yard score, which was awarded following a review.