Both teams had scoring chances within the first four minutes, as Gary Roberts tried to catch Bradford goalkeeper Ben Williams off his line from long range and James Hanson testing Tommy Lee at the other end.
Jon Stead then saw his shot saved, before the on-loan Huddersfield striker shot over the bar 10 minutes later.
Byron Harrison gave Chesterfield the lead in first-half stoppage time, directing the ball into the bottom right corner after Ritchie Humphreys assist.
Rory McArdle saw his shot saved from close range after the restart with Christopher Routis's shot going over the bar as Bradford searched for an equaliser.
