Malick Kone had a career-high 17 points and Myles Mack added 15 for Rutgers (3-1), which led the entire way but let an 18-point, first-half lead dwindle to one when D.J. Irving hit a 3-pointer with 8.2 seconds left.
Irving had 24 for Boston University, which has started 0-4 for the second straight year. Maurice Watson Jr. scored 16, John Papale 12 and Dom Morris 10.
After Irving pulled the Terriers to 77-76, Rutgers got the ball into the forecourt with 6.3 seconds left before Zach Chionuma was called for a fragrant foul on Jerome Seagears.
Seagears made both foul shots and, after Rutgers inbounded, Carter was fouled quickly and hit both with 5.7 seconds left to make it 81-76 and seal Rutgers third straight victory.