Fletcher ended a drought stretching back to August 2006 with an emphatic header from 12 yards in the 31st minute and he struck again late on, before Jack Lester made sure of Chesterfield's third successive league win.
Colin Woodthorpe had headed Bury level, but Chesterfield were the better side on the night and deserved to continue their impressive start to the season.
There was only one side in the game in the first half when Bury were over-run in midfield and the Spireites looked sharper all over the pitch.
The only surprise was that it took them until the 31st minute to break through, but once Fletcher had powered in a header from a right-wing cross it only looked a matter of time until the home side scored again.
Derek Niven brought a fine diving save out of goalkeeper Jim Provett with a 25-yard drive and Lester almost got the second after Jamie Lowry played him in on the right.
But the Shakers showed they still carried a threat when substitute Andy Mangan got through on the Chesterfield right, only for his low shot to be well saved by Barry Roche.
But the warning was not heeded by the home side and in the 76th minute Bury drew level.
A crossfield ball from the Bury right caught Chesterfield napping and Woodthorpe just beat Mangan to the cross to head low past Roche's left hand.
That stung Chesterfield to an instant response and within two minutes they were back in front.
Lowry swung the ball over from the right and Fletcher was again given too much space to plant a header inside the far post.
Any doubt about the result was removed in the 84th minute when Bury were punished again for more poor defending, as Lester was left unmarked and he brought the ball down to drive a low shot past Provett from 12 yards.