Middlesbrough dominated the first half, Robert Huth's early header over the bar signalling the visitors' intent.
Gareth Southgate's men nearly took the lead when Tuncay Sanli broke clear in the area but Carson was alert to make an excellent point-blank save, the ball rebounding to Mido, who fired narrowly wide.
Boro continued to press and Mohamed Shawky fired over from the edge of the area following a sweeping move.
Villa were guilty of repeated unforced errors and it was Knight's mistake that set up Downing for the goal.
The hosts finally had a sight of goal shortly after but Young screwed his shot well wide from 20 yards.
But Boro continued to threaten, Shawky firing a foot wide of Carson's right hand post from 25 yards, and Gary O'Neil's defence-splitting pass nearly setting up Tuncay.
Villa nearly equalised through Carew but his shot from six yards was cleared by the lively Mido with Mark Schwarzer beaten.
Downing, pursued by Knight, fires in the opening goal
Two half-time substitutions by Martin O'Neill, bringing on Marlon Harewood and Olof Mellberg, helped his side make a better fist of things in the second half.
But despite a Barry chip onto the roof of the net and a weak Young header directly at Schwarzer, they rarely looked like scoring until the penalty was awarded.
Boro could feel aggrieved as a shot by Villa's Young hit his Boro namesake on the elbow and referee Steve Bennett pointed to the spot.
Barry struck the penalty calmly to Schwarzer's right as the keeper went the wrong way.
Tempers began to fray towards the end as Boateng and Huth were booked for needless offences.
Aston Villa manager Martin O'Neill: "It was a hugely frustrating night and we just couldn't get going. We did not play well and did not deserve any more than the point.
"It was a particularly poor first half and it took us too long to get going.
"Our performance concerned me. We didn't create enough in the game and that is what disappointed me. We have to be a bit more inventive."
Middlesbrough manager Gareth Southgate: "I have seen the penalty incident again and our lad (Luke Young) has got his back to the play.
"The shot is hit from only a couple of yards away and I don't see the handball is anything but involuntary.
"We should have wrapped the game up but we are fighting for our lives and it was a very difficult decision to take."
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