As the race resumed after Monday's rest day, Welshman Geraint Thomas of Ineos Grenadiers retained the pink jersey with no further changes in the top five since Remco Evenepoel quit the race Sunday night with Covid.
The Ineos Grendiers leader said the weather was so bad he was simply happy to get through the day.
"I'm not taking anything for granted in this race, a lot can happen, especially with the weather. We'll take it day by day," said Thomas, who leads Primoz Roglic by two seconds while Tao Geoghegan Hart is third at five seconds.
Cort, who sports a handlebar moustache and also has wins on the Tour de France and the Vuelta a Espana, said the race was the most demanding he'd ever experienced.
"Today was such a hard day, one of the hardest stages I've done on a bike. At times I was sitting out there in the cold I was so confused I didn't know what was going on. My radio wasn't working, I think it got some water," said the 30-year-old.
"I can't remember doing a stage pushing all day like this for four or five hours, it's a very hard day."
A high-quality escape group featuring Alessandro De Marchi of Jayco-AlUla and Derek Gee of Israel Premier Tech got away and stubbornly refused to relent.
The peloton closed to within 40 seconds of the trio but dangerously slippery road surfaces causing constant spills forced a prudent approach to the Viareggio finish line.
The affable Dane who lit up the Tour of France in 2022 romping through Denmark in the king of the mountains jersey, left his sprint until last in a three-way cat and mouse at the line.
Another Dane, Mads Pedersen, won the sprint from the pink jersey group 51 seconds down while the rest of the bedraggled peloton came in 11 minutes adrift with Australian Jay Vine definitively out of the running after starting the day in the top 10.
The race conditions were brutal on the 151 riders, down from an initial 176, as heavy showers, gusts of blustery wind and temperatures as unseasonably low as three degrees Celsius (37 degrees Fahrenheit) on the stage's high point of 1527m on the Passo de Radicci.
The Giro's scheduling in May frequently makes it especially vulnerable to changeable weather.
Cort's win makes up some way for his teammate Alberto Bettiol being knocked down when someone from a team car ran out into the road towards an accident scene, without looking.
