Leading 76-72, the Explorers (6-3) made 8 of 12 free throws in the final 1:19 to hold off the Terriers (4-6), who closed to within a point on three occasions in the last 41 seconds.
D.J. Irving hit a free throw with 10 seconds left to make it 82-81, but missed the tying foul shot. Pettis hit his two free throws with 8 seconds left and BU's John Holland missed a 3-point attempt with 2 seconds to go.
La Salle trailed for most of the first half, falling behind 30-18 before closing to 38-35 at the break.
The lead changed nine times in the second half, with the Terriers holding their last advantage at 70-69 with 3:48 left.
Darryl Partin scored 20 points to lead BU, which shot 13 of 29 from beyond the arc.