The new-look Shakers saw their first game under new ownership spoiled by smash-and-grab goals from Jay O'Shea and Marc Richards.
The hosts enjoyed a string of opportunities and Tom Soares missed an absolute sitter, firing high and wide from Harrad's excellent cross.
O'Shea appealed for a penalty after going down inside the Bury box but the referee instead booked the Chesterfield winger for diving.
Bury then watched another chance go begging when Lee dived full stretch to keep out Nathan Cameron's powerful shot on the stroke of half-time.
Surprisingly, Chesterfield took the lead in the 50th minute when O'Shea cut in from the right flank to fire a left-foot shot into the far corner.
The Spireites sealed victory 16 minutes later with a devastating counter-attack, breaking downfield at pace for Richards to drive a low shot into the bottom corner.
Eoin Doyle wasted a chance for his side's third, seeing his shot cleared off the line in injury time.
- Sports & Recreation
- Shaun Harrad
- Marc Richards
- Tom Soares