Andre Nation added 12 points for the Dukes, who will make their first Colonial Athletic Association championship appearance since the 1996-97 season.
Third-seeded James Madison (19-14), which will take on top-seeded Northeastern on Monday, trailed by as many as 10 points but drew to within 57-56 with less than a minute to play.
Nation missed a potential game-winning layup at the 15 second mark and Delaware's Jarvis Threatt grabbed the rebound. Threatt attempted to move the ball upcourt but James Madison's full-court trap forced the guard to unintentionally dribble out of bounds with 11 seconds showing.
Moore drove the lane on the Dukes' next possession and drew the foul to set up his game-winning free throws.
Jamelle Hagins' 12 points led second-seeded Delaware (19-14).