Rovers played some excellent free-flowing football from start to finish, dominating possession and creating no fewer than 19 shots at goal.
Derby were always a danger on the break and only a smart low save from Jodie Mountain kept the scores level after 10 minutes.
Rovers eventually took the lead midway through the first half when Aisha Yallop’s attempted chip was blocked by the onrushing keeper, but Hardy was first to react and fired the ball into the unguarded net from 30 yards.
The visitors soon doubled their lead in similar circumstances, as a mistake by the Derby stopper was capitalised on again by Hardy, who lobbed the goalkeeper and retreating defenders from 35 yards out.
The second half couldn’t have started any better for Rovers, as Lorrie Doze’s through ball was pounced on by Hardy, who calmly claimed a well-deserved hat-trick.
Hardy scored her and Rovers’ fourth with a powerful half-volley, before Derby pulled one back with a header 15 minutes from time.
Rovers could have extended their lead in the closing stages, as Doze and Chandarana both went close after well-worked moves, but they were worthy winners and will now face either Arsenal, Aston Villa or Everton in the last four.
Rovers’ Under-17 head coach Matt Johnson said: “It’s a fantastic achievement to reach the semi-final of the National Youth Cup. I am looking forward to the draw and the opportunity to be the first Northern team to make the final.”
Rovers squad: Mountain, S. Hardy, Seed, Godfrey, Mallett, Musker, Brannigan, Doze, Adams, R. Hardy, Yallop, Williamson, Chandarana, Ratcliffe, Lambe.