Kosir had to make do with a win in the small final on Friday after losing to eventual winner Andreas Prommegger in the semi-finals however he wasn't to be denied second time out.
With two runs per round scheduled this time as opposed to one, Kosir beat Fischnaller in both as the Italian, who lost to Prommegger in the big final on Friday, was the runner-up again.
Fischnaller failed to finish the first run of the big final before losing the second by 1.37 seconds to Slovenian Kosir, whose best previous World Cup finish was third.
There was better news for Italy in the small final though as Aaron March beat off the charge of Russia's Vic Wild on both runs to take third place.
Meanwhile it was all change in the women's parallel slalom in Austria as Patrizia Kummer beat Julia Dujmovits in the big final and Svetlana Boldykova triumphed in the small final.
Amelie Kober led home Marion Kreiner and Claudia Riegler defeated Natalia Soboleva on Friday but it was last year's World Cup champion Kummer's turn to hog the headlines.
Much like the men's big final Dujmovits didn't finish run one before Kummer took run two by 0.34 seconds while Boldykova edged an all-Russian small final with Alena Zavarzina.
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- Andreas Prommegger