Derby won the tie on penalties after drawing 2-2 in normal time.
Goalscorer Harry Wilson, on-loan from Liverpool, said: "I know we are a good team, we wanted to come here and go toe-to-toe with United. The free-kick was too far out to curl it so I put my laces through it and tried to get movement on the ball."
Jack Marriott said: "We proved what a strong squad we have got, it shows what we are about. Penalties were nervy but we back our ability.
"Harry has got that technique on lock. We all had confidence in him. We are playing with confidence, we are together and we are a tight-knit group"