Scarborough were leading 1-0 in the first third, before Derby scored in the second third to set up a tense final 20 minutes.
The winning goal came with the last kick of the game via Liam Love, who had the confidence to chip the keeper after finding space.
Manager Luke Tinker said “The boys really took the game to Derby and stayed composed in the final third.
“It was a great team performance with players supporting each other on and off the ball. A big thank you to all the parents who got up at the crack of dawn to drive the boys down”.