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 tremendous 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.
The defeat for Dortmund is the team's second in the Bundesliga this season and follows up the 1-0 loss to Arsenal in midweek.
Wolfsburg made one change from the side that won at Eintracht Frankfurt last Saturday, with Daniel Caligiuri replacing the injured Diego.
Long-term absentee Bas Dost was named among the substitutes after recovering from an ankle injury.
The visitors were boosted by the return of defender Mats Hummels, who missed the Arsenal game with a thigh injury, while Jakub Blaszczykowski was ruled out with a gastrointestinal infection.
Dortmund coach Klopp had called for a reaction after Wednesday's Champions League setback, and he will have surely been pleased by his team's bright start to this game.
Hummels gave the hosts an early scare after five minutes when he found himself on the end of a whipped Pierre-Emerick Aubameyang cross from the right, but could not keep his header down.
Aubameyang was at the centre of things again four minutes later, breaking free of his marker inside the box before firing just wide.
Wolfsburg had chances of their own, though, and nearly took the lead just after 20 minutes through former Dortmund forward Ivan Perisic, who headed just wide.
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, the latter wasting a great opportunity 10 minutes into the second half.
Some great work from Henrikh Mkhitaryan saw the Armenian cut the ball back to the former Saint-Etienne striker, who scuffed his shot wide from 12 yards.
Dortmund were left to rue that miss just two minutes later when Rodriguez's teasing 35-yard free-kick from 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.
The goal shocked Dortmund, who came close to hauling themselves level 15 minutes from the end when Reus saw his free-kick come back off the post.
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.
The result leaves BVB four points behind Bayern at the top of the Bundesliga ahead of the sides' meeting following the international break.