Austrian Julia Dujmovits had moments earlier won the first women's title on the event's Olympic debut but it was Wild's second triumph for his adopted country after his giant slalom gold on Wednesday that brought the crowd to a crescendo.
Wild had looked down and out in the semifinals but staged a remarkable second run comeback to reach the final, where he edged giant slalom bronze medalist Zan Kosir in a thriller.
“Beyond believable,” said the 27-year-old Wild.
“When I came to the Olympics and showed up I had already won. To win the other day was the greatest feeling of my life. I can't believe it.”
Benjamin Karl lost to Wild in the semi but took bronze to give Austria their first two Extreme Park medals on the last day of action at the Caucasus Mountains venue.
The silver and bronze behind Dujmovits went to Germany's Anke Karstens and Amelie Kober, the latter winning the “small” final despite riding in a splint after injuring her elbow on Wednesday.
The Extreme Park program ended as it had started with a big crowd watching the qualification rounds bathed in bright sunshine under picture-perfect blue skies.
Both world champions, Slovenia's Rok Marguc and Ekaterina Tudegesheva of Russia, went out in the last 16, as did the women's parallel giant slalom champion in Sochi, Swiss Patrizia Kummer.
Also making an early exit was Russian Alena Zavarzina, who prompted Wild's nationality switch when they wed in 2011 and won bronze on the giant slalom course on Wednesday.
It looked like being a double disappointment in the slalom when Wild made a mistake five gates from the end of the opening run of the semifinal to fall more than a second behind Karl.
The slalom sometimes suffers from the need to give the riders a go on both sides of the course but the second run of this race lacked nothing in drama.
Wild burst out of the gate and was soon into the rhythm that had made him the standout slalom performer this week, closing the gap on Karl before leaning his body forward to slide across the line by 0.04 of a second.
After that, the final could have been an anti-climax but Kosir charged back after making a slight mistake halfway through the first run to keep matters tight.
Wild, though, was not going to let the second gold slip out of his hands and carved his way down the 325 metre slope and around the 28 gates to claim victory by just over a tenth of a second, sending the crowd into a frenzy.
“All those power drills I did in the summer, they really paid off. Nobody could keep up,” said Wild. “It has taken a lot of hard work, man. When everyone else in the summer is taking vacation, I am working hard.
”I train, I train, and it paid off.“