Ciro Immobile's superb strike 10 minutes after the interval put the Italians 5-3 up on aggregate and looked to have all-but sealed a last-four berth.
But in a week of stunning European comebacks, Salzburg added its own to the collection with a scintillating reversal to turn the tie on its head.
Munas Dabbur gave them hope through a deflected effort before Amadou Haidara kicked off the four minutes of madness with wonderful long-range drive after 72 minutes.
With momentum squarely on their side, the host then pulled level on aggregate, with Hwang Hee-Chan the latest Salzburg man to benefit from a deflection.
Stefan Lainer then had the final say with a back-post header to complete a remarkable turnaround and leave Italian hearts broken for the second time this week.