The Austrian side stretched its unbeaten record on the road in Europe to 10 matches, handing Dortmund manager Peter Stoger his first home defeat since taking charge in the process.
After being the better of the two sides in a scoreless first half, Salzburg struck first when Berisha converted a penalty following Omer Toprak's foul on Hwang Hee-chan.
The midfielder struck again to double his side's lead in the 56th minute, a vicious first-time strike from Stefan Lainer's cross putting the visitor in control.
Andre Schurrle bundled in from close range to half the deficit just after the hour, and Dortmund piled on the pressure in search of an equaliser.
However, it paid the price for a sluggish first hour that has left the Bundesliga club needing to score at least twice in the return fixture if it is to progress to the quarter-finals.
Marc Bartra's presence lifted the home crowd prior to kick-off, with the Spanish defender – who joined Real Betis in January after requesting a move back to his homeland – given a rousing reception on his return.
His old team-mates struggled in his presence, though, mustering just one on-target attempt – and that was a speculative 30-yard drive from Mahmoud Dahoud easily saved by Alexander Walke – before the break.
Salzburg, in contrast, were bright and busy and they were denied a deserved half-time lead by the woodwork, Hwang rattling the post with a right-footed effort from the edge of the penalty area in the final minute.
Yet the visitor was handed the chance to break the deadlock in the early stages of the second half.
Hwang managed to wriggle his way beyond marker Toprak wide on Salzburg's right, forcing the defender to desperately grab hold of his shirt and gift Berisha the chance to score from the spot.
If conceding first did not wake up a drowsy Dortmund, they were finally stunned into life when Berisha ran onto Lainer's clever pass from the right flank to thunder home via the underside of the bar.
Stoger sent on Christian Pulisic in response and the substitute wasted little time in making an impact, his low cross from the right allowing the onrushing Schurrle to force the ball over the line.
The goal gave Dortmund's comeback bid wings, yet the away side valiantly held firm in the heavy rain to make sure they will have a lead heading into the reverse fixture at the Red Bull Arena Salzburg.