Bradford, who had already disposed of top-flight Wigan Athletic and Arsenal, took the lead when Nahki Wells scored from close range in the first half before Rory McArdle doubled the advantage after the break.
Villa squandered several good chances with substitute Darren Bent guilty of arguably the worst miss, heading over an open goal in the second half before Andreas Weimann pulled a goal back late on for the visitors.
It proved to be a brief ray of light on a grim evening for Premier League strugglers Villa however, as Carl McHugh added gloss to the scoreline with two minutes remaining.
"You are playing against a class team, you need to be on your game," Bradford goalkeeper Matt Duke told Sky after a superb individual display.
"We've got an even bigger challenge ahead of us now."
The second leg at Villa Park takes place on January 22. Chelsea host Swansea City in the first leg of their semi-final on Wednesday.
Teams from the bottom tier of English professional football playing in cup finals is almost unheard of but buoyant Bradford will fancy their chances such is youthful Villa's poor form at the moment.
SHIPPED GOALS
Villa manager Paul Lambert made four changes to the team which beat Ipswich in the FA Cup on Saturday with Christian Benteke, Weimann and Gabriel Agbonlahor spearheading the visitors' attack.
They began the brighter and should have taken the lead after 10 minutes when Benteke found himself unmarked in the penalty area but could only direct his header wide from four metres.
Villa had shipped 17 goals in their last four league games and their leaky defence was exposed again after 19 minutes when a deflected shot from Zavon Hines fell kindly for Wells who calmly sidefooted the ball into the back of the net.
The goal acted as a confidence boost to the hosts, eighth in League Two, who got the upper hand in midfield and created a series of half chances as Villa looked short on ideas.
The visitors' best chance arrived just before the break when Charles N'Zogbia beat his man on the right flank and pulled it back for Agbonlahor who rifled in a shot from close range that Bradford keeper Duke did well to beat away.
The Premier League side looked brighter after the break and Duke was called into action early on, producing a series of good saves to deny first N'Zogbia then Benteke and Agbonlahor.
Striker Bent was a second-half substitute for Agbonlahor and within minutes of coming on, he wasted the visitors' best chance when he headed over from five metres after Duke parried a shot from N'Zogbia.
Cash-strapped Bradford, last in the top flight in 2001, looked to have taken a decisive advantage when McArdle powered home a header from a Gary Jones cross but Villa kept a toe-hold in the tie when Austrian Weimann beat Duke to the ball and poked home.
Their celebrations were short-lived as McHugh rose powerfully in the area to head home Bradford's third.
"Set pieces are a vital part of the game and
