Compatriot Mathilde Gremaud won silver, while Britain's Isabel Atkin took the bronze in a final that had confounded many pre-competition predictions.
Sochi gold medallist Dara Howell was eliminated in qualifying, while none of the top-10 of the World Cup standings got on the podium.
In glorious sunshine at Phoenix Snow Park, fans were treated to an entertaining final and it was Swiss pair Gremaud and Hoefflin who made the early running to lay down the two highest scores of the first round.
Norway duo of Johanne Killi and Tiril Sjaastad Christiansen both fell on the rails in first run. Sweden’s Emma Dahlstom, who qualified in first place with a 91.40 - the highest score of the day - also fell on the first kicker.
Further falls for Sjaastad Christiansen and Dahlstrom and a disappointing run for Killi in the second round meant the top three in the world were all struggling going into the final runs.
Hoefflin, currently 12 in the FIS World Cup, then raised the bar further as she improved her score to 91.20, taking the lead off her compatriot.
Gremaud, who admitted afterwards she had been suffering from a head injury in the build-up to qualification, went for the switch double 1080 on the final kicker of her last run to take back the gold, but fell going for glory.
Britain’s Atkins produced a clean third run to score 84.60 and move into third place. She then faced a nervous wait as the final three athletes took their final runs.
After neither of the highly-touted Norwegians could break into the top three with their final attempts, it was down to world championship silver medallist Dahlstrom in the final run of the afternoon.
However, the Swede couldn’t deliver and Switzerland got their first freestyle skiing gold since Evelyne Leu won the aerials in 2006.
"I think there were a lot of girls who could have won," Hoefflin told reporters afterwards.
"But one thing that is so great about women’s free skiing is it really is anyone’s to win."
"But I was really lucky in the finals because I didn’t have any wind in my runs, compared to some of the others. That helped me a lot."
Atkin, who is also a FIS World Championship bronze medallist, celebrated with friends and family after sealing the bronze medal. In doing so, Atkin won Britain’s first Winter Olympic medal on skis.
"I wasn’t aware that was the case until someone told me earlier in an interview,” said the 19-year-old.
"I was just stoked I got to ski with my team-mate Katie (Summerhayes) today."
"I wasn’t too concerned about the result (coming in the final) but I am so pleased to come away with the bronze."
It was a disappointing afternoon for many of the world’s top skiers.
Fifteen-year-old Jennie-Lee Burmansson, first in the world cup rankings, was aiming to become Sweden’s youngest ever winter Olympic medallist but could only manage an eighth place finish.
Birthday girl and Sochi silver medallist Devin Logan couldn’t repeat her feat from four years ago and finished in 10th.
(Reporting by Jack Tarrant; Editing by Amlan Chakraborty)
