Rutgers (12-11, 4-6) whittled the deficit to 64-60 with 2:38 remaining after switching to a full-court press midway through the second half, but the Scarlet Knights didn't get any closer the rest of the way.
Cotton hit two 3-pointers in 20 seconds, both on assists from Council, to give the Friars (13-10, 2-8) a 70-60 lead with 1:51 remaining. Cotton finished with 17 points.
The win snapped a four-game losing streak for the Friars, who had won just one of their previous nine games.
Providence began the game with a 7-1 run in the first 2:11, but the Scarlet Knights tied it 11-11 less than 4 minutes later.
Henton scored four straight points in the middle of the half to spark a 25-5 run that lasted nearly until the break. Coleman finished the scoring spree with an alley-oop dunk and free throw that made it 40-24 with 1:12 remaining. He finished with 15 points and seven rebounds.
Rutgers' Jerome Seagears charged the length of the court and hit a buzzer-beater from the lane to make it 40-26 at the break. It was only the third time this season that the Scarlet Knights had given up 40 points in the first half.
Council had 11 points and 10 assists by halftime, guaranteeing him his seventh double-double of the season. Henton, who was coming off a career-high 33-point performance in an 81-78 loss at South Florida on Sunday, also had 11 points in the half.
Rutgers cut into the lead and trimmed it to four points, 56-52, with 6:49 remaining in the second half.
Neither team was able to pull away Cotton's two knockout 3-pointers.
Before the game, Providence released a statement saying Sidiki Johnson, a 6-foot-9, 235-pound forward who had been suspended at Arizona on Nov. 21, will transfer to the school next season.
Johnson, who left the Arizona program 13 days after being suspended, played a total of 7 minutes in three games for the Wildcats this season and averaged 0.3 points and 0.7 rebounds per game.