The 25-year-old from Ottawa put down a fast run with a late start number to finish second behind Austrian Hannes Reichelt. Adrien Theaux of France was third.
Canadian skiers Erik Guay and John Kucera won men's world downhill titles in 2011 and 2009 respectively and Michael Janyk also won slalom bronze in 2009 in the bi-annual championship. Canada was shut out of the medals in Schladming, Austria, in 2013.
Cook has yet to finish in the top 10 in a World Cup race, but indicated he could handle the Birds of Prey course in December when he was 12th in super-G. He matched that career-best result the following week in a super-G in Val-Gardena, Italy.
Starting 28th, Cook crossed the finish line in a time of one minute 15.68 seconds behind Reichelt's time of 1:15.79.
"I pretty much lost my mind," Cook said. "I just screamed 'what?'"
"Obviously everyone is surprised because I'm not a 'name' on the circuit yet.
"I knew if I put a run together the podium was not a stretch and definitely possible. I was a little bit surprised to come down and see that I'm in second place for sure, but definitely not as surprised as everyone else who has no idea who I am."
His medal is a much-needed spark for the Canadian alpine team that has produced a pair of silver World Cup downhill medals this season.
Guay has yet to race this season and Olympic super-G bronze medallist Jan Hudec is not racing in Beaver Creek because of knee injuries.
Manny-Osborne Paradis of Invermere, B.C., arrived in Colorado bruised from a pair of crashes last month in Europe and did not finish Thursday's race. Morgan Pridy of Whistler, B.C., finished 22nd and Ben Thomsen of Invermere was 27th.
Former national-team skier and current television commentator Brian Stemmle was quoted in the Vancouver Sun earlier this week lamenting the state of the men's ski team heading into the world championship.
"Even they would agree. They suck right now," Stemmle told the Sun.
Cook said that comment wasn't on his mind Thursday, but he was happy to show the team has depth of talent.
"It's a nice little way to say 'see, we're actually doing all right,'" he said. "It's not something that went through my mind at all today except when it got brought up, but it was nice to say 'we lost our top guys for sure for now. They will be back, but we still have young guys who are doing well.'"
Cook grew up in Ottawa, but lives in Lac-Sainte-Marie, Que. His father Paul was in Beaver Creek to watch his son's triumph as was Cook's girlfriend Abby Ghent, who is a member of the U.S. ski team.
The Beaver Creek course is one of Cook's favourites because it's in North America and the "grippy" snow rewards risk-takers, he says. Cook was a little late getting to the start hut Thursday as there were only two racers to go when he arrived.
"The start was a little bit hectic. I thought I had a little bit more time," Cook said. "I really didn't have too much time to think about anything.
"I thought that I skied a little bit safe up top and on the bottom, I thought I skied really well and was aggressive and took it to the line."
Cook's medal was the 25th in the Canada's history at the world alpine ski championship.
A skier on the Canadian men's team who achieves a World Cup, world championship or an Olympic medal is designated a member of the "Canadian Cowboys" and receives a special belt buckle.
"As important as every World Cup medal is, our biggest tests are every two years at the world championships and every four years at the Olympics," said Paul Kristofic, Alpine Canada's vice-president of sport.
"For Dustin to be able to pull out his biggest career result and an epic result on the biggest stage is a testament to the strength he has and the future he has a ski racer."
American star Bode Miller, who tied with Hudec for Olympic super-G bronze, didn't finish after suffering a deep cut on his leg in a crash.
Miller hooked his left arm on a gate, spun backward, lost both skis and tumbled down the hill. He retrieved his gear and finished the race. It was Miller's first race since back surgery in November.
Norway's Aksel Lund Svindal, the 2013 world downhill champion and a two-time World Cup overall champion, raced for the first time this season Thursday and finished sixth. Svindal tore an Achilles' tendon while playing soccer in October.