A former world under-23 time trial champion and current Australian national champion, Durbridge beat Wiggins by one second on Grenoble's fast, flat, city centre course. Ukrainian Andriy Grivko was third.
Durbridge, who like Wiggins is a former world team pursuit gold medallist who has switched his focus fully to the road, said he had no idea he could outpace the Briton.
"I'm speechless, I wanted top five at best, I did not expect this at all," Durbridge said.
"To beat these guys that I've always looked up to, I mean Bradley Wiggins is such a class bike rider and I know he's in very good form. I really did not come here to get a result - it's amazing."
Durbridge said he had been fortunate with the weather, given it had rained intermittently after his early start time.
When Wiggins rolled down the starting ramp, the roads were still drying after the rain, making an all-out effort riskier.
"I was very lucky, but as sometimes that goes one way and sometimes that goes the other," the Orica-GreenEdge pro said.
"We made that right choice, it made for a long wait, but if I hadn't done that, this win might not have happened."
The first year Australian pro will now wear the leader's jersey on Monday's hilly stage from Seyssins to Saint Vallier, but says he is under no illusions when it comes to winning the race.
"My targets were today and Thursday's (53 kms) time trial which will be the longest I've ever ridden. I'm not here to do general classification. Just winning the stage and leading the race tomorrow is amazing anyway," he said.
The 2011 Tour de France winner Cadel Evans finished a solid ninth place but 2010 Tour champion Andy Schleck of Luxembourg, making his Criterium du Dauphine debut, was well off the pace in 102nd.
"I've never been with the strongest when I race the Tour de Suisse but then I'm always stronger in July," Schleck told French television. "I'm going okay for now."
The race finishes next Sunday in Chatel.