The weather was foggy at the top of the slopes, which cleared towards the bottom of the course, while the snow conditions appeared more challenging at the finish. Those difficult snow conditions took most of the favourites and leaders from the first run out of the picture as Strasser, who was eighth after the first run, held on.
It was Austria's PyeongChang 2018 bronze medallist Michael Matt, who finished fourth overall, who clocked the fastest second run starting fifth of the 30 skiers, showing just how much the slopes deteriorated. However, despite clocking a time some eight tenths slower than Matt, Strasser's strong first run was enough to overtake Matt by around half a second.
Matt's teammates Manuel Feller and Marco Schwarz completed the podium behind Strasser, with first-run leader Clément Noel only seventh and slalom red bib Henrik Kristoffersen tied for 10th.
It was Schwarz's seventh World Cup slalom podium without a victory in the discipline, with only Paolo De Chiesa (12 podiums without a win) ahead of him on that list.
An early error from Alexis Pinturault on his second round saw him finish tied for 18th. However, the Frenchman still took enough points from the race to overtake Aleksander Aamodt Kilde – who did not compete in the Croatian capital – at the top of the overall World Cup Crystal Globe standings.
"I got a little bit emotional, I did not really realise… normally we have a big crowd here cheering for us and now it's just us, a small skiing family," Strasser told FIS after the race.
"Mixed feelings, but it's just an emotional victory for winning my first slalom event," the German – whose last win came in the Stockholm City Event in January 2017 – added.
With his victory, he became the first German to win a men's World Cup slalom race since 2017, when Felix Neureuther triumphed in Levi, Finland.
The skiers now move on to Adelboden, Switzerland, for a slalom and two giant slalom races.