Play-off candidates Norwich had crushed into a seemingly commanding lead but a dramatic finale saw former Manchester United youngster Ben Muirhead and summer signing Michael Branch pull of a great escape.
The cash-strapped Bantams, who made wholesale changes in the summer, had been cursing their luck until the duo's late double brought them just reward for their sweat and effort.
Former Everton and Wolves man Branch showed glimpses of his quality in the closing stages of the first half and it took a full-length save from Norwich keeper Robert Green to push away his 38th minute strike.
A Branch snapshot flew wide soon after but Bradford's good work was undone by a rash challenge from Paul Evans after a long ball caught him napping.
As ex-Wimbledon man Damien Francis darted through Evans tugged at his shirt and despite the protests Mark Rivers stepped up to send a low right-foot penalty just beyond keeper Mark Paston.
The strike was a body blow to Bradford, but they responded well and Green had to scramble to tip over a looping shot from the dangerous Branch.
Within two minutes of the second half the Canaries had doubled their tally as they punished Bradford's ball-watching defence.
Francis was again the spark but when David Nielsen's angled shot was saved by Paston, Bradford should have cleared the danger.
Instead Neilson was able to pick out Clint Easton who fired home their second from 12 yards.
Bradford were given a boost when Branch was bundled over by Malky Mackay but the penalty chance was wasted by last season's top scorer Andy Gray.
The Scottish international struck the inside of the post with his right foot effort and his follow up flew wide of the mark.
Bradford's bad luck continued when Dean Windass' goalbound volley was blocked by Adam Drury's face and an Evans free-kick also clipped a post.
But Muirhead ignited a superb revival when he fired home Bradford's lifeline six minutes from time.
This time luck was on their side as his left-foot drive rebounded off a post and Green's head to give the Bantams hope.
Neilson saw a chance to secure the three points go begging when he blazed wide, but Branch made no mistake with his last-gasp stunner.
The Norwich defenders were frozen to the spot as Evans' hopeful long ball landed in the box and Branch took full advantage to lash home a priceless debut goal deep into injury time.