The pair scored four of the Tigers' seven points in the extra period, with Philmore hitting two free throws with 1:17 to go and Dupree scoring on a layup with 13 seconds left. La Salle's Earl Pettis missed a 3-pointer with 1 second on the clock.
Towson coach Pat Kennedy called the victory Towson's biggest since he took over the program seven years ago.
The Tigers (4-7), who led by 12 in the first half and 43-37 at halftime, didn't let the Explorers pull ahead until Pettis' free throw with 4:30 left gave La Salle a 72-71 lead.
The Explorers (6-7) built a five-point cushion, but Philmore's 3-pointer with 2 seconds left in regulation tide it at 86.
Jerrell Williams paced La Salle with a career-high 32 points and 13 rebounds.