Jones put the Bantams in front after nine minutes when his corner from the left squeezed in at the front post.
James Hanson should have extended the visitors' lead only to miss with a header from close-range, while Cody McDonald squandered Gillingham's best chance in the second-half as they slumped to the bottom of the table.
Bradford striker Nahki Wells drew a save from Gillingham goalkeeper Stuart Nelson before poor defending allowed Jones' near-post corner to find the back of the net.
Only a reaction save from Bradford keeper Jon McLaughlin prevented Gills captain Adam Barrett from equalising in the 24th minute.
Hanson should have extended Bradford's lead on the half-hour mark when unmarked six yards out, but put his header wide.
Gillingham's best chance came in the 51st minute when Adebayo Akinfenwa attempted an acrobatic shot and the ball fell to McDonald four yards out, but he blazed his shot over the crossbar.