Toarlyn Fitzpatrick chipped in 13 points and Augustus Gilchrist 11 for South Florida (13-9, 6-3 Big East), which has already surpassed its overall (10) and conference (three) victory totals from last season.
Robertson had four points and Jawanza Poland connected on a go-ahead jumper during an 8-0 run to put South Florida up 55-53 midway through the second half.
After Henton converted three free throws to cut the Providence deficit to two, 80-78, South Florida 's Victor Rudd Jr. hit 1 of 2 from the line with 5 seconds left.
Providence could do no better than play to a 39-all tie at halftime despite Henton scoring 24 points. The freshman forward entered with a career-best of 21 points, coming against Boston College on Dec. 8.
Vincent Council had five points early on in the second half as Providence took a 46-41 lead. Henton didn't take his first shot of the second half, an in-close basket that gave the Friars a 51-45 advantage, until the 7-minute mark.
The Bulls grabbed a 69-60 lead on Robertson's 3 with 4 1/2 minutes to go. After Providence got within 78-75, Rudd secured the South Florida win on two late free throws.
Henton had a three-point play, hit a 3 and Bryce Cotton added a long-range jumper to put Providence ahead 36-29 with 4:15 left in the first half. Cotton had 16 points overall.
Anderson scored two of his 12 first-half points on a game-tying layup 12 seconds before halftime.
Council finished with 10 points. The junior guard had 26 points, 10 rebounds and nine assists in the Friars' previous game, an 86-74 loss at Pittsburgh on Wednesday.Click here for more info and to get the inside scoop!