Burnley defender Clark Carlisle had cancelled out Villa substitute Emile Heskey's opener with just a minute of normal time remaining but Downing curled home a fine effort after 96 minutes.
Villa had Marc Albrighton sent off four minutes after the goal - for a professional foul - but last season's beaten finalists held on to reach the last eight.
Albrighton will now miss Sunday's derby against Birmingham at Villa Park.
Burnley keeper Lee Grant was called into meaningful action for the first time after 16 minutes when Ashley Young's first-time pass gave Stephen Ireland a run at goal.
But the former Manchester City player opted to shoot first time and Grant was able to save to his right.
Nigel Reo-Cooker should then have put Villa ahead after 24 minutes when Young's low cross left him with only Grant to beat six yards out.
But the midfielder side-footed his shot and the keeper was able to save. The rebound broke to Barry Bannan but he too was denied by Grant.
Dean Marney became the first player to be booked after 28 minutes when he brought down Young just outside the box after he had raced onto Ireland's pass.
Bannan made good contact with a swerving 25-yard effort early in the second half but Grant was well positioned to gather the ball low to his right.
Burnley skipper Graham Alexander than made a fine challenge on Ireland as he prepared to shoot after being found in space by Reo-Coker.
Burnley came close to taking the lead after 55 minutes.
Tyrone Mears' quick free-kick released Martin Paterson who turned inside Beye before firing a low cross-shot just past the far post.
It then needed a double save from Friedel to keep Villa on level terms.
First he fingertipped away a shot from Paterson and then reacted quickly to keep out the rebound header from Rodriguez.
The Villa keeper also had to be alert to turn over a looping header from Rodriguez.
Downing had the chance to make a quickfire impact when Young's pass sent him clear of the Burnley defence but he was well off target after lofting the ball over Grant.
Villa finally broke the deadlock after 86 minutes through Heskey after Young's low centre had presented him with a close-range tap-in he could not miss.
However Burnley forced the game into extra-time with a minute remaining, Ross Wallace supplying the cross from the right and Carlisle climbing above the Villa defence to head past Friedel.
Villa restored their lead after six minutes of the first half of extra-time, though. Downing was allowed time to check back onto his favoured left foot and, although Grant got a hand to the ball, he could not stop his fierce shot from entering the net.
Villa were reduced to 10 men after 110 minutes when Albrighton brought down Burnley substitute Wade Elliott as he raced through on goal.
Friedel made a fine save to keep out Wallace's free-kick and then Burnley appealed in vain for a penalty after Eric Lichaj appeared to handle.
The Villa keeper also did well to turn aside Elliott's close-range flick.
In the last minute Friedel kept out Mears' shot and Burnley substitute Chris Iwelumo had a header cleared off the line by Carlos Cuellar.