Bauer, who had been widely expected to vie with World Cup leader Northug for several medals at the Vancouver Games, took bronze.
"I ran an amazing race from the start to the finish and I felt very good," a beaming Cologna said after surging across the finish line in overcast but dry conditions. "I am so happy. With this victory, my childhood dream came true.
"I had a good push at the end, but I tried not to push too hard at the start," added the Swiss, who led the World Cup standings after the 2008-09 season.
Cologna earned Switzerland its second gold medal of the day, following Didier Defago's victory in the men's downhill, by taking control over the last 7.5 km of Whistler's twisting trails.
He had trailed Swede Marcus Hellner by three seconds at the halfway point of the race but found another gear over the second circuit before skating to the finish line in style.
Piller Cottrer, who won Olympic gold in the 4x10km at the 2006 Turin Games, was delighted with his podium finish.
"Getting silver behind such a great athlete is gold for me," the police officer said. "To jump on the podium means my (preparation) programme was excellent."
Northug, a winner of three golds at last year's world championships in Liberec, Czech Republic, finished well down the field in 41st place after a disappointing run.
Scotland's Andrew Musgrave, in his maiden Olympic appearance, came home 55th, clocking 36:32.4 minutes and finishing 19 places ahead of compatriot Andrew Young.
And while that was 14 places better than his last World Cup outing in Canmore earlier this month, 19-year-old Musgrave admitted he was feeling the pace.
"My shape hasn't been that good coming into the Olympics but it did feel a little bit better out there," he said.
"Today shows that I have improved but it is nowhere near my best. I haven't got a clue why I am not in good shape."
Young admitted his gameplan was all wrong.
"It wasn't too great - I worked on some tactics and stuck to them but I don't think that worked out," the 17-year-old said.
"I don't know what it is maybe I have got a cold or something but I just couldn't ski as fast as I wanted to and that is a little bit annoying."
Place of birth: Mustair, Switzerland
Residence: Davos Platz, Switzerland
No previous Olympic results:
Champion, 2008/09 World Cup, overall
Runner-up, 2008/09 World Cup, distance
Background: Cologna started the sport when he was 12. The 2009 Tour de Ski winner is currently third in this season's World Cup overall standings.