Shiffrin was second after the first run and was given quite a challenge by overall World Cup leader Tina Maze, who had a blazing second run to take a large lead just before the American was to ski.
But Shiffrin kept her composure in a run that was even faster, setting a time that first-run leader Hansdotter couldn't match.
Hansdotter was still ahead of the American at the first two intermediate times but wasn't as fast in the final section and finished 0.29 seconds behind for second place. Maze was third, 0.52 behind.
Lindsey Vonn skipped the race, having taken a break from the World Cup to fully recover from an intestinal illness.
Shiffrin became the youngest women's World Cup winner since Lara Gut of Switzerland won a super-G in St. Moritz in 2008, also at 17. She already had two podium finishes in slalom, but the impressive victory is likely to create even higher expectations on the teenager. And she's not quite sure she's ready for that.
"I don't really like the success," she said. "I like to sleep at night and I like to hang around the hotel room with my mom. I'm afraid there might be a little hype with this. But I'll take it. This is what I love to do."