Adam Smith made it three goals in two games with the only goal of the game at Warminster Road, as Chesterfield extended their run of unbeaten games to seven.
Chesterfield kept the pressure on Hull City and Darlington at the top of the League with a 1-0 win over Bradford City thanks to Smith's second half strike.
The Spireites failed to display the kind of football they have produced in previous weeks and went in with the scores level at the break.
But A Smith broke the deadlock with a 68th minute strike from 18 yards out, which beat the Bantams goalkeeper.
Dave Bentley's side picked up another vital three points which now leaves the pressure on at the top, with the Spireites having games in hand over the top two clubs.
Chesterfield (4-4-2): Malak, Granger, Clare, Holmes, Burns, Adcock, Darikwa, S Smith (Watkis 68) (Tingay 84), Clay, A Smith, Bowery
Subs Not Used: Needham, Webb, Burrow