"It was my dream since I was a little kid. We have to plan some good party when I get home," she said at the finish.
The 20-year-old was fastest by just over half a second on a first-run course set by her father Amir.
As the course became rutted and bumpy on the second run, other late starters struggled.
Ljutic went last as leader but showed no sign of nerves. Apart from one mid-course stumble, she rocketed down the course to win by a massive 1.75sec.
"I felt all over the place," said Ljutic, who had finished on the podium five times since 2023.
"I really didn't know if I am fast enough, so I try to push all he way down. I really wanted to take this chance, and I did."
German Lena Duerr, second after the first run, struggled but clung onto the runners-up spot, one tenth of a second ahead of Austrian Katharina Liensberger, who jumped from seventh to third with the second-fastest second run.
Swiss racer Camille Rast, the leader in both the overall and slalom World Cup standings, was third in the first run but slipped to fourth.
Ljutic climbed to second overall and third in the slalom standings, behind Duerr.
