The most successful World Cup skier of all time dominated the first run and did not hold back in the second either, as she beat Rast by a combined 1.5 seconds.
Shiffrin was the overwhelming favourite after winning seven of eight World Cup slaloms this season, but the lack of medals after two events, along with a blank in Beijing four years ago, had ramped up the pressure.
The 2014 Olympic slalom champion made clear it was back to business right from the outset, leading by 0.82 after the first run.
Germany's Lena Duerr was the sole skier to come within a second of her time in that run, but her hopes evaporated in a split second on the decisive leg after she straddled the first gate out of the hut.
Sweden's Cornelia Oehlund had been third after the first run, albeit a second off the pace, but she too was unable to convert that into a medal after breaking a ski pole and failing to finish.
Shiffrin saw what happened and then put in a flawless run. As the crowd cheered, it took what seemed an age before she raised her arm and a ski pole to acknowledge what she had achieved.
She appeared to be crying as she embraced her mother, Eileen.
The 30-year-old, who had spoken on Instagram earlier in the week of "narratives built on a limited understanding of what this sport truly demands", had finished fourth in the team combined with Johnson and 11th in the giant.
Had she also skied out on Wednesday or made a costly error, there would have been plenty of critics ready to air their opinions.
"She receives a lot of press, good and bad, and she's very good at handling it. But it gets to somebody at some point if people are only saying bad things about you," said Moltzan, who went from 28th after the first run to eighth.
"We're just human at the end of the day. And she's just a human. Mistakes happen. You don't have the perfect run, and we get ripped apart by people that sit on the couch, and that sucks."
Rast's silver was her first Olympic medal, while Swenn Larsson's bronze made her the oldest female Olympic Alpine medallist after Italian Federica Brignone, a double champion in Cortina.
"It's my last Olympics... for sure I am not here in four years, so it feels extra special to put it together today," said the Swede.