BVB dominated their Austrian hosts during the first half of the third qualifying round, first-leg tie and led through Jonas Hofmann.
Wolfsberger understandably protested after an offside decision went in favour of Marcos Reus during the build-up, but Hofmann - who endured an injury-plagued spell on loan with Mainz last term - made no mistake when the chance fell his way.
The Bundesliga outfit allowed their levels to drop as the second half wore on and Thomas Zundel passed up a glorious chance as Wolfsberger's confidence grew.
Joachim Standfest and substitute Tadej Trdina had efforts cleared off the line for Didi Kuhbauer's side, who remain in the tie ahead of next week's second leg at Signal Iduna Park.
Pierre-Emerick Aubameyang headed off target from Hofmann's second-minute cross as Dortmund immediately established early control.
Wolfsberger's high defensive line looked set to cause them problems during the early stages and so it proved in the 14th minute when Dortmund opened the scoring in controversial fashion.
Clearly in an offside position as Ilkay Gundogan played the ball forwards, Reus was adjudged not to have interfered with play as he left it for Henrikh Mkhitaryan, who steamed towards goal from the left and cut the ball across to leave Hofmann with a simple finish.
Mkhitaryan again found space down the Wolfsberger right after 28 minutes but his low cross to the far post was too far ahead of the onrushing Aubameyang.
Reus teed up Hofmann to have an attempt blocked before Dortmund goalkeeper Roman Burki - preferred to Roman Weidenfeller, who Tuchel has confirmed will play in the second leg - dealt unconvincingly with a Manuel Seidl free-kick in a rare moment of encouragement for the hosts.
The majority of the action remained in vicinity of the Wolfsberger penalty area when the game resumed, with Alexander Kofler saving excellently from Hofmann’s low strike before Aubameyang provided a second wayward headed attempt.
With an hour played, the home team masterfully cut Dortmund open but, following a neat turn and pass from Seidl, Thomas Zundel blazed wastefully over.
Apparently buoyed by that attack, Christopher Wernitznig bundled beyond Dortmund full-back Lukasz Piszczek and cut a dangerous centre across the face of goal. Piszczek recovered well and was on hand to block Standfest's effort from Zundel's cutback.
The next Dortmund man to perform heroics on his own goal-line was Mkhitaryan, when Trdina met Seidl's right-wing corner.
The offside flag belatedly came to Wolfsberger's aid in the 76th minute, when Mkhitaryan had his celebrations correctly cut short, and Tuchel's men preserved their slender advantage despite another Seidl corner causing further havoc, Burki excelling himself to thwart Boris Huttenbrenner and Silvio.
