The Yorkshire outfit bounced back from two successive home defeats with a gritty display and almost snatched the three points in a frantic finish.
Skipper David Wetherall was inches away from diverting home an 80th minute chance but only a late tip over from Davison denied Paul Dickov an even later winner.
Bradford's teenage hot shot Danny Forrest almost embarrassed Leicester's Gerry Taggart after just five minutes when he pounced on his mistake but the 18-year-old dragged his shot wide.
The youngster, who scored a stunning overhead kick against Walsall, then shot straight at keeper Ian Walker before the Foxes started to turn up the heat.
A series of corners tested the Bradford defence to the limit and Davison had to save brilliantly at the feet of Trevor Benjamin on the edge of his six-yard box.
Just after the half hour the lively Dickov almost broke the deadlock with a neat near-post flick but Davison was again equal to the task.
The Leicester man, who had already had a 25-yard strike ruled out for offside, was a constant threat and the Bantams rode their luck as the half wore on.
Foxes skipper Matt Elliott rattled their bar with a 16-yard volley after another corner was only half cleared and striker Brian Deane had another goal scrubbed out because of an offside flag.
The second half was slow to get going but there was plenty of action in the final quarter.
Referee George Cain waved away three Bradford penalty appeals and was also unimpressed when Dickov went tumbling in the box.
The Foxes haven't won at Bradford since 1937 and Davison's finger-tip save from Dickov's 88th minute shot made sure that that run continued.