The League Two outfit takes a two-goal advantage into the second leg at Villa Park after the defender's late header capped off a glorious night for Phil Parkinson's side.
Nakhi Wells opened the scoring with an opportunistic goal in the first half, before poor defending allowed Rory McArdle to double the home side’s lead after the break.
Andreas Weimann lifted the ball beyond Bradford goalkeeper Matt Duke to slice the deficit in half, but Carl McHugh headed past Shay Given in the closing stages to give Paul Lambert’s side a two-goal hole to overcome at Villa Park.
Villa was punished in the 19th minute after failing to fully clear a corner, as Hines’ 20-yard effort was deflected into the path of Wells, who stroked the ball past Given, handed the start in place of American Brad Guzan, the Villans' No. 1 keeper.
Villa began the second half on the ascendancy, and Christian Benteke was foiled by the Bradford ‘keeper, who kept out the Belgian’s close range diving header before denying Gabriel Agbonlahor, who connected well with a Barry Bannan cross.
With little more than 30 minutes to play, Darrent Bent was brought on in place of Agbonlahor as Lambert pressed for an equalizer, but the striker fired over the bar from a narrow angle with one of his first few touches of the ball.
The striker missed a golden opportunity to equalize when Duke saved a shot from Charles N’Zogbia and the ball looped up in front of an open goal, but the England international headed over.
The Premier League side was made to pay for the host of missed chances for the second time as woeful defending from another corner allowed Wells to keep an attack alive for Jones, whose cross was met by a glancing header from McArdle at the near post, which beat Given.
The visitors scored eight minutes from time as Weimann raced onto a loose ball and lifted it over Duke to put Villa in a bit less of a disadvantageous position, but McHugh’s header in the final minute of normal time restored Bradford’s two-goal advantage, taking the League Two side within 90 minutes of a Capital One Cup final at Wembley Stadium.