The Tigers (1-2) trailed 7-6 in the fourth quarter when Jordan Mynatt blocked Jeremiah McGeough's punt and Overstreet recovered it in the end zone. Connor Frazier connected with Andre Dessenberg for the 2-point conversion.
After a late Delaware State (0-3) drive stalled at the 24, Joseph put the game away with an 81-yard punt return touchdown with 2:54 remaining.
Darius Victor broke open a 50-yard run as Towson milked the clock. He finished with 131 yards rushing.
The Hornets' lone score came on a 13-yard pass from Gilbert Rivera to Malik Golson with 7 seconds left in the second quarter to make it 7-3. Sam Hurwitz's second field goal of the day pulled Towson within one in the third quarter.