Gary Roberts gave Chesterfield the lead before Mayor was dismissed and goals from Jay O'Shea, Marc Richards and debutant Dan Gardner completed an impressive display.
Roberts had two decent chances before he opened the scoring in the 16th minute when he drove the ball into the roof of the net at the back post after an O'Shea cross had skimmed off the head of a defender.
Bury almost equalised in the 22nd minute when Anton Forrester was allowed space to shoot from 20 yards but it was too close to Tommy Lee.
Bury had Danny Mayor sent off for a lunge on Matty Brown just before half-time and Chesterfield strengthened their grip on the game when O'Shea cut in from the left and beat Brian Jensen with an unstoppable shot from 25 yards shortly after the hour mark.
The game was over in the 76th minute when Richards turned the ball in from close range after Roberts had headed on a cross from Armand Gnanduillet, and Gardner broke away in the 86th minute to beat Jensen and wrap up the scoring.