The trio held off a group that included Mark Cavendish to set up a three-way sprint, with Cort timing his move perfectly to take the win.
Cort has won two stages at the Tour de France and six at the Vuelta a Espana, and completed the Grand Tour stage sweep with his first Giro victory on his second appearance in the race.
"I'm extremely happy to complete my trilogy, winning a stage of the Giro d'Italia after the Tour de France and La Vuelta," said Cort, who rides for EF Education-EasyPost. "But it's been one of the hardest days I've spent on the bike.
"Firstly I had to get back to the front. Then my radio wasn't working because of the rain so I was never sure where the chasing peloton was. But we kept pushing hard. It was worth it. It's hard to believe that it happened though."
Canada's Gee (Israel-PremierTech) and Italian De Marchi (Jayco-AlUla) finished second and third, respectively.
Geraint Thomas maintained his two-second lead over Primoz Roglic after both riders finished in the peloton, 51 seconds behind Cort. Tao Geoghegan Hart is third overall, five seconds behind his Ineos Grenadiers team mate Thomas.
Thomas inherited the leader's maglia rosa jersey after pre-race favourite Remco Evenepoel had to withdraw after testing positive for COVID-19.
"It was nice but obviously not ideal conditions to be leading the Giro," Thomas said. "It was pretty cold on top of the hill and crazy in the descent.
"I'm happy with getting through this stage alright. In this race anything can happen, especially with the weather."
Wednesday's stage 11 is a 219km ride from Camaiore to Tortona, the longest stage of this year's race.
