The Austrian side were already 2-0 ahead when Soriano collected a stray ball just inside his own half, took a touch to get the ball out of his feet and sent a stunning strike looping over goalkeeper Jasper Cillessen.
Salzburg captain Soriano opened the scoring from the penalty spot and was also involved in Sadio Mane's 21st-minute effort before capping proceedings with his memorable effort.
Andre Ramalho wasted an early opportunity to put the visitors ahead but it did not take long for them to get on the scoresheet, Joel Veltman conceding a 12th-minute penalty by lunging two-footed at Alan.
Soriano stepped up for the spot-kick and sent Cillessen the wrong way with a calm finish.
Ajax were cut open again six minutes later as Soriano laid off for Kevin Kampl, whose precision pass landed was well controlled by Mane.
The Senegal international rounded Cillessen with ease and rolled into an open net.
The best was yet to come, though.
Ricardo van Rijhn inadvertently played a part, diverting an innocuous loose ball into Soriano's path near the centre circle.
The 28-year-old spotted the goalkeeper off his line and, after rolling the ball no more than a yard into the Ajax half, lashed a perfect shot high into the air and just over the back-tracking Cillessen.
His reaction was understated given the execution, but his team-mates rightly mobbed their skipper.
Bojan Krkic and Viktor Fischer combined to test Salzburg just before the break, but Ajax only truly threatened in the closing seconds of the tie.
Bojan went close in added time before Siem de Jong drew a fine save out of Peter Gulacsi seconds before the final whistle.