Bergmann, who came ninth in Bad Gastein a year ago for his only previous top-10 result, defeated Andreas Prommegger of Austria in the final of the single-run format.
Aaron March of Italy finished third by beating Sebastian Kislinger of Austria.
Sylvain Dufour remained in the lead of the slalom standings with 1,290.0 points. Prommegger trails the Frenchman by 230 points in second place while Bergmann climbed to fourth with 1,016.7.
In the women's competition, Kummer defeated Ester Ledecka of Czech Republic for her first win of the season and ninth overall. Marion Kreiner of Austria took third ahead of Ekaterina Ilyukhina of Russia.
Kummer went to the top of the slalom standings with 1,400 points, 150 clear of second-place Ledecka. Caroline Calve of Gatineau, Que., who led before the event, was beaten in qualifying and dropped to fourth.
Several more favourites, including Olympic champions Jasey-Jay Anderson of Lac-Superieur, Que., and Nicolien Sauerbreij, and world champion Isabella Laboeck, also failed to reach the top 16 in qualifying earlier Friday and missed out on the finals.
The event was moved from Jauerling because of a lack of snow in Lower Austria. Bad Gastein hosts another parallel slalom on Sunday.