The Bantams took just four minutes to go in front when a neat passing move saw Jones assist Gillead and the on-loan Newcastle United winger scored from ten yards.
Stuart McCall's high-flying League One outfit doubled their lead on the stroke of half time - less than 60 seconds after Chesterfield's Kristian Dennis' effort was disallowed for offside.
Straightaway, Bradford broke and Charlie Wyke played Jones through to fire past advancing goalkeeper Joe Anyon.
The home side had the ball in the net again in first-half stoppage time but Wyke's header was disallowed for pushing.
McCall's side had chances to extend the lead further in the second half but three superb saves from Anyon kept the score respectful