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.
With a trio a top recruits from Philadelphia, the Penn State basketball program is primed to finally turn a corner.
Maryland point guard Melo Trimble, considered a lock to leave the program before struggling in the second half last season, has withdrawn from the NBA draft and will return for his junior season.
Louisville's Chinanu Onuaku, who recently underwent a medical procedure to correct a heart rhythm issue detected during the combine, will keep his name in the NBA draft pool.