On the final climb up Mount Cermis, Cologna won with a comfortable margin of 1 minute, 26.5 seconds ahead ofMartin Johnsrud Sundby of Norway.
Alex Harvey of Canada finished third, 1:30.6 behind.
Cologna joined Poland's Justyna Kowalczyk as the only four-time winners of the Tour.
Sundby recorded the fastest time of the stage to move up from sixth overall at the start of the day.
Weng began the day trailing teammate Ingvild Flugstad Oestberg by 1.8 seconds but pulled away and won by 48.5 seconds with the fastest time of the stage in the 9-kilometer free pursuit.
"I knew when it got steep that was when I wanted to make my move," Weng said.
Jessica Diggins finished third, 2:23.2 behind, to become the first American on the podium in the 12-year history of the week-long event.