The revelation of the season, who last month celebrated his first World Cup victory in the super-G in Wengen, completed his run in one minute and 40.68 seconds to clinch his first medal in a major event.
Austria's Vincent Kriechmayr - a two-time world medallist - had to settle for a silver, finishing 0.24 seconds behind.
"I wasn't nervous, I was really happy to ski here, the slope was great," said a smiling Von Allmen.
"It was a run really on the limits but at the end it went really well... I tried to feel no pressure, I had nothing to lose, just try to perform."
Von Allmen, a triple silver medallist at 2022 Alpine Junior World Ski Championships, was 0.31 seconds faster than Monney.
The Swiss duo increased the medal tally for their country to four, level with hosts Austria.
"It's amazing, one day it's Odermatt, the other day two other athletes from the Swiss team. It's really fun that we can push each other like this," Von Allmen added.
The 2023 champion Odermatt, who added men's super-G gold to his career collection on Friday, finished fifth.
"It was a tight race," Odermatt said. "I knew I had to take the risk today, I really did. I could ski, full gas, but I certainly did two, three mistakes and that, on a downhill like here, you cannot win."
The 2022 Olympic gold medallist Odermatt praised his younger teammates for their brilliant performance.
"(They) did an incredible job. If I don't win, they do, so that's good," he said.
"(Von Allmen) is a crazy guy. He doesn't think much. He just goes all the way full speed and, if everything goes well like today, he can win every race. He's going to be a big rival for the future."