From practically the first attacking move of the match the 20-year-old on loan from Blackburn Rovers ran on to a pass from Darren Ferguson and placed his angled shot across Pirates goalkeeper Scott Shearer from the left edge of the box.
It was just the start Wrexham needed after a run of four defeats but they looked tired from the effort of playing four matches in eight days.
Once in front, they were content to keep a tight defence and block any holes that their visitors might exploit.
Simon Spender missed a second opening for the home side, shooting wide, and a free-kick from Lee McEvilly cleared the crossbar.
A deep cross from Spender was just too high for Derbyshire to connect as he went up for it at the back post.
Rovers could not find a decent shot on goal in the opening half but they pushed up later and created pressure without giving young home goalkeeper Michael Jones any real saves to make.
He was drafted into the senior side at short notice after first choice stopper Mike Ingham went to hospital for an X-ray on an injured wrist.
Steve Elliott headed wide from a corner floated in by Rovers substitute Ali Gibb and from Rovers' next half-chance Stuart Campbell shot high into the stand.
A shot from Craig Disley was deflected for a corner off the Wrexham goal line and when another strike from Chris Caruthers deflected off Richard Walker it flew straight into Jones' arms.
Caruthers fired a late shot off target before Chris Lines saw a shot blocked on the line.