Gut, who was third after the opening run, won on Friday with a total time of two minutes 2.51 seconds with Austria's Eva-Maria Brem the runner-up by .10 of a second and Italy's Federica Brignone third on 2:02.85.
"It's a tough course," Gut said.
"There are no easy gates," she said.
"It's good to be on the podium here."
Reigning world and Olympic slalom champion Shiffrin, who led after the opening run, was three gates from victory but did not finish, handing Gut her second career victory in the discipline after Solden, Austria, in 2013.
"I didn't feel good throughout the race," Shiffrin said.
In the World Cup overall and discipline standings, Brignone, who won this season's giant slalom opener at Solden, remained in first place by 160-150 over Gut with Brem third on 112.
Three-time defending World Cup slalom champion Shiffrin had been second at Solden and would have taken the World Cup lead with a triumph.
Instead, the 20-year-old American stands sixth on 80.
She was third last season in the giant slalom.
"Mikaela was skiing awesome," Gut said.
"To win the overall, you have to ski fast. I focused on that. It's too early to talk about the overall. We will see in March."
Gut struggled in giant slalom last season, finishing 24th in the discipline and sparking a major revamp.
"Being fourth in Solden I realized I could compete again against the best," Gut said.
"I fixed since March my problem in giant slalom. I changed everything, new skis, new skiman, Didier Cuche and Daniel Albrecht came to
help — the puzzle is coming into place."
US ski star Lindsey Vonn crashed out in the first run. The four-time World Cup champion was running seventh when she faltered, the third of 18 skiers who failed to finish the opening run.
"My left ski came off," Vonn said. "To be honest I am a bit perplexed. It wasn't even bumpy. It never happened to me in a race before. To happen in Aspen is really disappointing."
Vonn, 31, moved on to next week's speed events in Canada, a course where she has dominated in past seasons.
"Next week it's Lake Louise," she said.
"I am going to move from today and focus on that."