Gut-Behrami posted a time of one minute 14.91 seconds to finish 0.35 seconds ahead of Norway's Kajsa Vickhoff Lie, with in-form Italian Federica Brignone completing the podium.
The win extends Gut-Behrami's lead over Brignone in the super-G standings, which the Swiss skier also won last season along with the giant slalom, and also cuts the Italian's overall lead.
Gut-Behrami is now just 70 points off Brignone in the overall competition, despite the Italian recording five World Cup wins this season, including the downhill on Saturday.
"I've been struggling a little bit, I had to get the confidence back, the way to push on the skis again, and I made mistakes," Gut-Behrami said.
"I'm happy that today everything came together."
Vonn, who had failed to finish her previous two races, was 1.40 seconds adrift of the winner as the 40-year-old four-times overall World Cup champion continues her comeback after coming out of retirement in November having quit the sport in 2019.
The next World Cup race is Thursday's slalom in Courchevel, where American Mikaela Shiffrin is set to return from injury and resume her hunt for a 100th World Cup win.