The 24-year-old -- who had finished fourth in the previous two super-G's -- was two tenths of a second faster than Vincent Kriechmayr of Austria. Swiss racer Alexis Monney filled out the podium, 0.24sec off the pace.
Moeller's first World Cup triumph followed that in Saturday's downhill of Monney, who said it had been "a crazy weekend".
Moeller said he had not changed anything in his racing style.
"I always try to just ski the way I know I can and that paid off today," said Moeller.
"I did not expect at all to win today but I had two good Super-G races before this.
"I knew my speed was good and the skis felt good, boots, everything.
"So I knew I could just ski the way I know and that is a really, really good feeling."
Moeller said that, while he felt in good form, victory was far from his mind given the strength of the Swiss challenge.
"I had a pretty good feeling today but to come out with a win, possibly, I did not expect that," he said.
"The Swiss guys are always there.
"It is cool to keep them behind me for at least one day.
"I will try to do that the whole season."
Monney's compatriot Marco Odermatt never threatened to make it a hat-trick of super-G wins at Bormio, finishing fifth, 45 hundredths slower than the victor.
Odermatt consoled himself by saying the main thing was he had emerged from the weekend unscathed, having narrowly avoided crashing in Saturday's downhill.
He admitted he had never been at the races on Sunday.
"This was a tough race for me, I did not have a good feeling from the starting gate," said the 27-year-old three-time overall World Cup champion.
"I lost a lot of time in one section and was unable to find the speed at the bottom of the course to claw back the time."
The Swiss pair fared better than teammate, Gino Caveziel, who appeared to suffer a leg injury after he crashed and was taken away by helicopter.
Kriechmayer was delighted to make the top three -- his best result this term -- but said his ageing legs, he is 33, were finding it hard to compete with the younger generation.
"I am looking forward to the next races," he said.
"The new generation is pushing hard, they are always on the limit, if you are not skiing up to your limits from the top to the bottom, you can't beat them."
Caveziel's mishap followed that of French skier Cyprien Sarrazin, who underwent surgery for cranial injuries suffered in a harrowing crash on Friday.
The 30-year-old Sarrazin was airlifted to hospital and went under the knife to treat a subdural haematoma after he slammed into the snow and slid into protective netting while training for the races.
