The Capital One Cup finalists, 3-2 down from Thursday's game at Valley Parade, recorded their first ever win at the Pirelli Stadium courtesy of Nahki Wells' double either side of James Hanson's goal.
Jacques Maghoma levelled up the tie from the penalty spot but Burton could not get themselves back on terms again after Wells had struck his 25th goal of the season - and his fifth against Albion this campaign.
Burton started brightly and Maghoma created an opening for himself after just 24 seconds. The former Tottenham trainee showed good feet to turn on the edge of the box but his left-foot shot went wide.
Shortly afterwards Alex MacDonald's long throw into the penalty area found Calvin Zola who rolled Andrew Davies and got a shot away, but the two-goal hero from Valley Parade fired into the side-netting.
MacDonald then saw his far-post header from Damien McCrory's left-wing cross blocked just in front of the line by a wall of Bradford defenders.
It was from another Burton attack that the visitors opened the scoring after 27 minutes.
Jon McLaughlin made a routine save from McCrory's free-kick and quickly distributed the ball to James Meredith, who knocked it long. Albion centre-half Marcus Holness tried to head back to his goalkeeper but did not get enough on it and Wells nipped in to poke past the onrushing Stuart Tomlinson from inside the box.
Wells thought he had won a penalty shortly afterwards when he was flattened by Holness. Referee Graham Scott pointed to the spot but to Burton's relief the offside flag had already been raised and after consulting with his assistant Scott changed his decision.
Hanson put Bradford ahead on aggregate for the first time after 50 minutes.
Wells' persistence enabled him to poke the ball into the path of Hanson and the big striker drilled a right-foot shot into the bottom left-hand corner of the net from 20 yards out.
However, five minutes later the hosts levelled up the tie. Maghoma turned Garry Thompson out on the left touchline and powered towards the penalty area with the Bantams midfielder in tow.
Thompson made a last-ditch challenge which referee Scott adjudged to be inside the box and Maghoma got up to score from the spot.
But Burton were level for only two minutes. McLaughlin's long punt forward was headed down by Hanson and Wells spun his marker to flick a left-foot shot past Tomlinson from six yards out.
Immediately at the other end, McLaughlin preserved Bradford's advantage with a save to deny Robbie Weir. And the Bantams keeper kept his side ahead again nine minutes from time, blocking substitute Billy Kee's point-blank header with his chest.
Bradford had to survive some late Burton pressure but they did and they will return to Wembley for the final on May 18.