Tobias Werner put Augsburg in front 10 minutes in and, for at least the first half-an-hour, there was only one team in the game.
The hosts failed to add a second, though, and were punished by two goals in seven minutes, with Maximilian Arnold and Luiz Gustavo turning the game on its head and earning Dieter Hecking's men all three points.
Augsburg came flying out of the blocks, testing Diego Benaglio early on through Hong Jeong-ho's effort before Halil Altintop was denied by a fine one-handed save.
Wolfsburg's resistance was broken when Werner scored, providing the finish after first Altintop's shot was saved by Benaglio and then Sascha Molders' effort from the rebound was blocked.
Molders only just managed to get out of the way of Werner's shot.
Andre Hahn missed the chance to make it two, while a cross flashed inches in front of Werner as Augsburg looked to a second a second.
Altintop pulled a shot wide before, out of nowhere, Arnold levelled.
His 25-yard effort whizzed past Alexander Manninger, who could only watch the ball fly into the left-hand corner of his net.
And it was now Augsburg's defending which proved not up to scratch as they left Gustavo completely free to head in Ricardo Rodriguez's free-kick from six yards out to complete the turnaround.
Augsburg offered little by way of response after the break, seemingly lacking the energy to do so after such a bright opening.
Diego went the closest to scoring at the other end, but Manninger guarded his near post well to keep the Brazilian out as Wolfsburg held on to their first away win of the season.