Striker Ashley Barnes put Burnley ahead just before half-time by rounding goalkeeper Emiliano Martinez in the wrong direction during the penalty shoot-out after Burnley were awarded a spot-kick for a foul by Villa on Maxwel Cornet.
The visitors didn't have to wait long for an equalizer when Emiliano Buendia made amends for a penalty by scoring the equalizer in the 48th minute with a fine first time shot from inside the box.
Villa dominated the final period but Burnley goalkeeper Nick Pope was too tough to lift Mike Jackson's side out of the bottom three.
Burnley are currently 17th on 35 points or level with Leeds but have a much superior goal difference ahead of their last game at home to Newcastle United on Sunday. Villa itself remains in the order of 14.