The Austrian side, unbeaten since August, secured a shock 2-1 lead in Germany last week through Valon Berisha's double, but goalkeepers Alexander Walke and Roman Burki were equally tested in this return stalemate despite Dortmund's need for at least two goals - a tally they never came close to achieving.
Stoger's gameplan made it easy for the hosts, Dortmund's attempts to play the game entirely in Salzburg's half opening up space for the energetic Hwang Hee-chan to run free.
Burki twice denied the 22-year-old forward and then produced a wonderful stop to thwart Xaver Schlager as the visitors somehow reached the break level on the night and still in with a chance of progression.
Their improvement after the restart was minimal, though, Dortmund managing to keep Salzburg at bay but doing little to muster the requisite attacking response.
Dortmund's season is far from over as they remain well placed to seal Champions League qualification, but this shock exit for one of the tournament favourites will come as a blow to Stoger's hopes of staying in the dugout beyond the end of the campaign.
Dortmund made a particularly laboured start and might have fallen further behind in the tie within just six minutes as poor defending allowed Hwang in, Burki blocking his clipped finish.
The visitors assumed complete control of possession but another error at the back then let Hwang run clear once more, the South Korea international this time pushed wide and drilling hard and low for Burki to save again.
And Burki continued to single-handedly keep Dortmund in the contest, diving to his right to brilliantly tip Schlager's curler around the post.
A spectacular punch from Berisha's teasing cross-cum-shot was next for Burki to deal with, Dortmund's aimless possession only inviting snappy Salzburg tackles as the opening 45 minutes passed without event for Walke in the home goal.
1 - This is only the second time this season that #BVB have had just one shot in the first half (the other game was the DFB Cup game at FC Bayern). Harmless.— OptaFranz (@OptaFranz) March 15, 2018
Powerful efforts from half-time substitute Maximilian Philipp and Marcel Schmelzer worked Walke after the restart, before the goalkeeper was forced into action again when the latter stooped to head on target.
Walke came up with a more difficult stop to keep a stumbling Alexander Isak out, then parrying from Philipp and watching on as Stefan Lainer stepped in front of a goal-bound strike.
Salzburg did grow a little nervy but were never troubled, the scale of the task in the closing stages too much for Dortmund as Walke earned a clean sheet.
And the hosts could have themselves stolen victory at the last as Munas Dabbur cracked the crossbar after Fredrik Gulbrandsen led a breakaway.