Darby's winner five minutes from the end of extra-time at Valley Parade on Tuesday night ensured City reached the last 16 of the Capital One Cup for the first time in 24 years.
The all-League Two third-round tie looked set for a penalty shootout when Darby - a close-season signing from Liverpool - picked up the ball 30 yards from goal and beat diving Burton goalkeeper Stuart Tomlinson with a low right-foot shot - his first goal for the club.
Earlier, two goals in the last six minutes by Bradford's Bermuda international striker Nahki Wells sent the tie into extra-time after the visitors had led 2-0. Burton went ahead after 18 minutes when Jack Dyer whipped a low cross in from the right and Billy Kee scored with a superb first-time shot from 15 yards and increased their lead 11 minutes later.
Goalkeeper Matt Duke turned away Lee Bell's shot and Aaron Webster scored with a header from Bell's corner. Bradford skipper Gary Jones headed the ball away from under the crossbar, but the linesman ruled it had crossed the line.
Bradford came close to scoring a minute before half-time when Stuart Tomlinson dived full-length to keep out defender Rory McArdle's header, while striker Garry Thompson wasted a good opportunity when he was left with only the keeper to beat.
The visitors endured several narrow escapes as Bradford dominated the second half and the home side earned their reward when left-back James Meredith threaded the ball through to Wells to score with a left-foot shot into the corner. Wells then equalised a minute from the end of normal time, turning the ball into the net from close range after the Burton defence failed to clear Jones' free-kick.
|Team Name||Bradford City||Burton Albion|
|Shots On Target||9||3|
|Shots Off Target||8||3|