The 2026 Olympic team combined bronze medallist beat Italian-born Lara Colturi, who skis for Albania, by the slimmest of margins - 0.01 of a second after two runs down the Rettenbach glacier slope - with Germany's Emma Aicher third.
The victory from fifth after the first run also made Moltzan the first woman to win with Van Deer skis, the brand founded by Austria's retired eight-time overall World Cup winner Marcel Hirscher.
"It feels like a fairytale right now," said the Minnesota-born skier.
"It doesn't really feel real, but we've put in a lot of hard work this summer developing the ski, and it feels like this is kind of a pinnacle, that it's paying off and we did all the right things. I can't imagine a better start."
The podium was the 11th of Moltzan's career, six of them second places including one in Soelden a year ago when Austrian Julia Scheib was a first-time winner.
Scheib did not finish the first run this time.
World Cup overall champion Shiffrin led by 0.14 seconds after the first run, with Sweden's Sara Hector second and Switzerland's Camille Rast third, but conditions changed for the later starters in the second leg with falling snow, flat light and grey skies.
The 31-year-old US great, chasing a record-extending 111th World Cup win, has not won a giant slalom since December 2023 but has shifted her focus this season towards that discipline and super-G, and will race fewer slaloms.
Saturday's race also marked the World Cup return of Slovakian Petra Vlhova, who had been out injured since January 2024. The 2022 Olympic slalom champion and 2021 overall World Cup winner finished 23rd.
Italian Federica Brignone, the Olympic champion in giant slalom and super-G on home snow in Cortina in February, was absent as she continues her recovery from surgery in August.