The Friars (8-2) shot well from 3-point range the entire game, led by Cotton, who went 5 for 11. Fortune hit 3 of 6 3-pointers.
The Friars sported green uniforms with "Sandy Hook" printed on the backs in honor of the shooting victims from Sandy Hook Elementary School in Connecticut on Friday.
"I wanted to do something to honor those victims," said coach Ed Cooley. "It made me feel really, really special that that was able to happen."
Freshman Kris Dunn made his collegiate debut, setting the Providence record for most assists in a debut with 13. Transfer Sidiki Johnson also joined the injury-ridden Friars lineup, coming off the bench with four points.
After playing all nine games this season with seven or fewer players, Cooley was happy to have Dunn and Johnson in the lineup.
"Kris was able to get other guys shots, and he created offense with his speed and athleticism," Cooley said. "I thought Sidiki was good. We've got to teach him a couple more things, but he was a physical presence."
Dunn helped share the ball-handling load with Fortune and Cotton, who have split the responsibility so far this season.
"It's always better when we have more people who can somewhat take the load off," Cotton said.
The Providence victory was marked by runs at the start of each half. The Friars used an early 17-2 spurt to take a 26-16 lead, and came out of the locker room with a 27-4 run in the second to seal the victory.
Providence also had a very strong defensive effort, with 48 rebounds, 11 steals and four blocks, three from Johnson.
Providence begins a three-game road trip on Saturday afternoon against Boston College.
"I'm just really concerned about how we're going to guard people as the competition continues to increase," Cooley said. "If you're going to be a good team, at the end of the day, you've got to win on the road."
Colgate, which has lost five games in a row, moves on to face St. Francis (N.Y.) on Saturday.
Former Kentucky guard Ed Davender, the only Wildcat with at least 1,500 points and 400 assists, has died. He was 49.
Guard Naz Mitrou-Long, whose 2015-16 season ended in December due to hip problems, has been granted a hardship waiver for the coming season, Iowa State said.
As the Sooners look to life without superstar Buddy Hield and his sidekicks, it's clear their unseasoned players will need to grow up fast.