Gut notched up her second success of the season and 21st overall with a time of 1min 23.24sec to beat Liechtenstein's Tina Weirather by 0.13s with Italian Elena Curtoni third at 1.24.
Home favourite Tessa Worley, winner of the last two giant slaloms, came in fifth.
Gut, the defending Crystal Globe champion, was making up for failure to finish in Saturday's downhill and the combined slalom 24 hours earlier.
The big loser of the day was Sofia Goggia, the Italian who could have taken control of the overall standings only to be disqualified.
Meanwhile in Switzerland, Slovenian teenager Domen Prevc maintained his explosive form with his fourth success of the men's World Cup ski jumping season at Engelberg on Sunday.
The younger brother of Peter Prevc extended his lead in the overall standings with a record 144-metre first jump on the Swiss slope to deny Poland's double Olympic champion Kamil Stoch.
Austrian Stefan Kraft came in third in this seventh event of the season.
Whilst it was all smiles for Domen, his 24-year-old brother Peter produced his worst performance since March 2014 by finishing a distant 33rd place.