Martin Gritton grabbed the winner with a neat curling finish in the 53rd minute after being played in by the lively Jack Lester.
The Bees looked jaded after three tough games in a week and the closest they came was when leading scorer Charlie MacDonald headed straight at Chesterfield goalkeeper Tommy Lee.
The home side’s striker Jordan Rhodes also missed two good chances in the space of a minute before limping off injured midway through the second-half.
The Spireites looked dangerous on the break and Drew Talbot should have wrapped it up on the hour when he rounded goalkeeper Ben Hamer only to hit the upright.
Talbot forced a fingertip save from Hamer minutes later before wasting a gilt-edged chance to make sure of the points in the 79th minute when he blazed over from close range.