Free-kicks from Marco Reus and Ricardo Rodriguez either side of half-time saw the game finely poised at 1-1 before Olic sent a stunning curled effort past Roman Weidenfeller to earn Wolfsburg all three points.
Reus struck another set-piece against the post late on, but the home side held on to secure their fourth straight league win.
Defeat for Dortmund is the team's second in the Bundesliga this season and second in the space of four days following a 1-0 loss against Arsenal in midweek.
Dortmund coach Jurgen Klopp had called for a reaction after Wednesday's UEFA Champions League setback, and he will have surely been pleased by his team's bright start to this game.
Dieter Hecking's men began to take the game to Dortmund as the half wore on. First, Slobodan Medojevic volleyed wide before Patrick Ochs had a long-range effort tipped over the bar.
Neven Subotic was carried off on a stretcher just before the break after landing awkwardly, but Klopp's worries were soon eased when Reus stepped up to curl a 25-yard free-kick into the top corner of the net on the stroke of half-time.
The goal predictably boosted Dortmund's confidence, with Reus and Aubameyang both looking lively, with the latter wasting a great opportunity 10 minutes into the second half.
Dortmund were left to rue that miss just two minutes later when Rodriguez's teasing 35-yard free-kick form the right eluded everyone before bouncing past Weidenfeller.
Three minutes later things got worse for Klopp's side when Olic received the ball on the right flank before cutting inside and sending a vicious left-footed strike in off the post from just outside the box.
Wolfsburg's nerves were tested late on with Robert Lewandowski having several penalty appeals turned down, although the home side could have extended their lead at the death when Perisic raced away from the Dortmund defence before firing agonisingly wide.
