The England winger settled the fourth-round tie with his powerful shot after 96 minutes. Downing was allowed time to check back onto his favoured left foot and, although Lee Grant got a hand to the ball, he could not stop his fierce shot from entering the net.
Burnley defender Clarke Carlisle had cancelled out Villa substitute Emile Heskey's opener with just a minute of normal time remaining.
Marc Albrighton had released Ashley Young down the right and his low centre was side-footed home from close range by Heskey. But with a minute remaining Burnley forced the game into extra-time.
Ross Wallace supplied the cross from the right and Carlisle climbed above the Villa defence to head past Brad Friedel.
Villa had Albrighton sent off four minutes after Downing had scored - for a professional foul - but last season's beaten finalists held on to reach the last eight.