Dortmund started brightly with Andre Schurrle blazing an early shot over the crossbar, but Salzburg hit straight back with Amadou Haidara firing a long-range effort marginally off target.
Both teams were having their moments in either final third, but clear-cut chances were few and far between until Mahmoud Dahoud was denied by Alexander Walke from distance.
Five minutes before half time, Salzburg's Hwang Hee-chan saw a shot blocked by the legs of goalkeeper Roman Burki, before Munas Dabbur had a header blocked by a defender.
Dortmund endured a nightmare start to the second half when Omer Toprak was penalised for shirt-pulling inside the box, allowing Berisha to step and guide his penalty past Burki to make it 1-0 to the visitors.
Salzburg soon made the most of a promising start by adding a crucial second away goal to their tally, this time as Berisha returned again to bag a brace when he latched on to a ball from Stefan Lainer in order to send a memorable shot past the Dortmund stopper.
The home side were not found wanting for a lack of effort, though, as they hit back in the 62nd minute through Schurrle's quick thinking to turn the ball home from close range.