Scott's running jumper with 7:13 remaining ended a Miami field-goal drought of 8:40 and stopped a 15-2 Scarlett Knights' run that cut into a 19-point Hurricanes lead.
Myles Mack's 3-pointer from the corner with 9:29 remaining capped the run and got the Knights to within six, 48-42.
Miami secured the win as Scott scored five points and Grant converted four free throws in the final 3:01.
Florida transfer Kenny Kadji scored 13 points for the Hurricanes (2-0).
The Hurricanes expanded a 35-25 halftime lead, outscoring the Knights 11-2 in the first 4:42 of the second half. Kadji keyed the run with five points, including a three-point play for a 45-27 lead with 15:53 left to play.
Miami finished 10 of 23 from 3-point range and Rutgers shot 9 of 23 from behind the arc. But the Knights only took six free throw attempts and failed to convert a shot.
The Knights rallied from a 16-point deficit with a 10-2 spurt late in the first half. Mike Poole's 3-pointer with 4:02 left capped the run and cut Miami's lead to 31-23.
The Hurricanes ran off to a 14-4 lead in the first 4:53. Scott and Kenny Kadji scored two field goals each during the spurt.
Consecutive 3-pointers from Scott and freshman Shane Larkin increased the Hurricanes' lead to 20-6 with 12:52 remaining. The 3-pointers were part of an 8:30 stretch when all six Miami field goals came from behind the arc. Larkin, son of former Cincinnati Reds shortstop Barry Larkin, played in his first collegiate game Tuesday.
Grant's 3-pointer with 6:37 left ended the string of long-distance baskets and gave the Hurricanes their biggest lead of the half, 29-13.