The game was goalless at the Bob Lucas Stadium which forced the game to spot kicks - Dartford missing all three.
Weymouth enjoyed a spell of dominance in the opening exchanges before Dartford took the initiative - Darren McQueen going closest, smartly getting his shot away as he fell but Calum Ward was equal to it.
After the break, Noor Husin's 20-yard strike was hit well but sailed high over the goal.
Substitute Ben Thomson then jinked his way into the box, beating three players before getting a low shot away but Sesay got a strong hand to it.
Jake Jebb dragged a shot wide two minutes from time but the final whistle went and penalties were needed.
Tom Whelan, Jake McCarthy and Jaidon Anthony all scored their kicks but Jebb, McQueen and Husin all fired over the Terras bar.
Weymouth: Ward, Camp, Harfield, Ngalo, McCarthy, McQuoid, Anthony, Wakefield, Robinson, Brooks, Hobson Subs: Childs, Zubar, Whelan, Thomson, Murray, Odubade
Dartford: Sesay, Wynter, Hill, Bonner, Braham-Barrett, Hyde, Husin, Jebb, Romain, Sheringham, McQueen Subs: Edwards, Marsh-Brown, Noble, Wanadio, Ming, Briggs