The Austrian finished 0.43 seconds ahead of first-round leader Clement Noel of France, while Feller's compatriot Marco Schwarz rounded off the podium.
Feller produced a stunning final run on a difficult course, which was a late replacement for the coronavirus-affected original leg in Wengen.
But on a day where incomplete runs were the norm, his uniquely aggressive style allowed him to snatch a late lead from last week’s winner in Adelboden Schwarz.
There were scenes of jubilation after for the Tyrol native, who tasted success for the first time after five previous podium finishes.
The fact his first victory came on home snow made it all the sweeter.
Feller triumphed on a track named after Austrian ski legend Hermann Maier.
"Right now [I feel it's] unbelievable," the 28-year-old told FIS after. "I have to say I was a little bit nervous before the race today. I tried to make a solid first run, in the second I knew the material was working really well.
"I didn't even feel on the limit, I felt like everything was just working perfectly. I want to thank all the guys from Flachau. When I was skiing I felt like having a crowd next to the slope. Right now it's like a miracle!"
Feller's victory means he now stands atop the slalom leaderboard, with Schwarz one point away in second and Linus Strasser of Germany in third.
However, with five different winners from five races, there is still everything to play for as we approach the second half of the season.
There will be little rest for the maiden winner, who is back in action tomorrow in the second men’s slalom race.