Lewis Chalmers nearly fired Aldershot in front on seven minutes. Louie Soares' deep cross was cleared to the midfielder on the edge of the area but his first-time shot was well-saved by Rhys Evans.
In the opening period Aldershot looked most likely to score, but it was Bradford that broke the deadlock on ten minutes from a well-worked corner routine.
Paul Heckingbottom's cross to the edge of the area was dummied by Joe Colbeck and the unmarked Paul McLaren slotted the ball coolly inside the left post.
On 17 minutes Evans again came to Bradford's rescue, rapidly retreating to tip Marvin Morgan's cute lob over the bar.
In the closing minutes, Lee Bullock miscued a certain goal into the arms of Nikki
Bull from six yards and, in stoppage time, John Grant was denied by another good Evans save having cut in from the left channel.
Ben Harding blasted Aldershot back into the game on 48 minutes with a rasping volley from the edge of the area. Moments later a last-ditch tackle from Soares denied Joe Colbeck.
On the hour mark the game went into over-drive. Anthony Charles was adjudged to have brought down Joe Colbeck and Nikki Bull dived low to thwart Peter Thorne's spot-kick.
Moments later Aldershot went in front when Morgan stabbed home Soares' cross. Three minutes later Bradford levelled, when Lee Bullock reacted quickest to Graeme Lee's flick inside the penalty area.
The game continued to build with more heart-stopping moments and on 82 minutes, Lee scrambled Kirk Hudson's cross off the goal-line.
Bradford thought they had grabbed the winner on 84 minutes after a terrible mix-up between Bull and Rhys Day.
But the drama didn't end there. On 86 minutes Davies smashed the winning goal from 28 yards against a team who nine years ago to the week were playing West Ham in the Premiership.