Andreas Weimann gave Villa a valuable away goal in the 83rd but Carl McHugh restored the more comfortable two-goal cushion with a powerful header in the 89th.
Villa can still turn things around at home, but is facing a much tougher task than expected after wasting a number of scoring opportunities against the League Two club. The second leg is at Villa Park on Jan. 22.
"A draw would have been great -- 3-1 is absolutely amazing," Wells said. "We're buzzing. Ahead two goals, we just need to stay resilient."
Bradford's only major trophy is the 1911 FA Cup title, and it has struggled both financially and on the pitch since a brief stint in the Premier League between 1999-2001.
Villa dominated early but were repeatedly thwarted throughout the game by goalkeeper Matt Duke, while Christian Benteke had two headers just high early on.
