John Bostock fouled Nicky Ajose with 12 minutes remaining and Morgan dispatched the spot-kick in front of 2,800 Owls fans who had seen their team squander several first-half chances.
Wednesday dominated much of the first half without finding a finish as Ryan Lowe, Reda Johnson and Lewis Buxton all went close before Lowe had convincing claims for a penalty turned down. However, Chesterfield had their moments with Stephen Bywater having to tip a 20-yard drive from Liam Ridehalgh over the crossbar.
The home side came into the game after half-time and Bywater made another good save to deny Drew Talbot in the 64th minute before Morgan was presented with the chance to make the breakthrough from the penalty spot.
The Owls almost responded immediately but Tommy Lee clawed away a Rob Jones header to leave Gary Megson's side regretting those early missed chances.
A third straight league defeat for the first time since March last year meant Wednesday missed the chance to reclaim second place in the table while the win saw Chesterfield, temporarily at least, move to within a single point of safety.