Collins, who scored on a layup, was fouled on the play, and made the ensuing free throw to make it 67-65 with 12.7 seconds remaining. He finished with seven points and 10 assists.
It was the first Big East Conference win for South Florida (8-7, 1-1), which had opened league play on Wednesday, losing 60-57 to No. 9 Connecticut.
The Bulls rallied in the final minute after Rutgers took a one point lead. After South Florida got a 3-pointer from Shaun Noriega, Eli Carter scored in the lane to give Rutgers a 63-62 lead with 1:24 left.
Carter had 23 points for Rutgers (8-6, 0-1), which was coming off an 85-83 double overtime win over No. 10 Florida on Thursday night. He missed a potential game-tying jumper in the final seconds.
Carter had a career-high 31 points in the victory over Florida. Rutgers dropped to 2-15 in Big East openers.
Rutgers started the game off slow, missing its first eight shots, falling behind 6-0. Carter stopped the dry spell on a 3 with 15 minutes remaining in the first half.
Augustus Gilchrist scored eight points to pace South Florida to a 29-22 halftime lead. The Bulls, keyed by Noriega's long-range jumper, scored the first half's final seven points. South Florida made just 4 of 9 free-throw attempts during the opening 20 minutes. Rutgers converted its two chances.
The Scarlet Knights opened the second half on a 7-2 run as Gilvydas Birutascored four points and Carter connected for a 3 to get Rutgers within 31-29.
Anderson's 3-point play put South Florida ahead 45-40 midway through the second half.
Carter made a layup and added a free throw to get the Scarlet Knights even at 55-all with 5 1/2 minutes remaining.