Alves tried his luck from long range 11 minutes before the interval, and Blades' keeper Kenny was forced to punch away his powerful shot.
The Brazilian turned creator after 42 minutes when he played in Luke Young, but he provided a defender's finish and Kenny saved with comfort.
Sheffield United held Middlesbrough at arm's length in the early stages of the second half, with only Alves threatening to break the deadlock with another shot that Kenny saved low down.
Given Middlesbrough's need for firepower, it was a surprise when Alves was taken off with 18 minutes left and replaced by Tuncay Sanli - a decision that was not well received by the home supporters.
James Beattie had been a peripheral figure, but almost snatched victory for Sheffield United in injury time when presented with a chance at the near post, but he shot wide under pressure from Boro keeper Mark Schwarzer.
It was the last clear chance in normal time, and Downing tested Kenny six minutes into extra time as he forced him to save with his feet - the keeper recovering to save well from Tuncay at his near post.
The game had degenerated into a mediocre affair, but the deadlock was broken in truly bizarre fashion with six minutes remaining.
Mido's shot was deflected on to an upright, but as Kenny tried to retrieve the rebound he could only turn the ball over his own line.
Both sides had chances in a frantic final few minutes, with Sheffield United having strong penalty appeals waved away when Emanuel Pogatetz appeared to handle.
• Middlesbrough manager Gareth Southgate:
"We had plenty of chances to win it, but the longer it went on the more it looked like it was going to penalties.
"But we have shown a lot of heart and character this season and we showed that again. Everyone left in the last eight of the FA Cup will think they are in with a great chance.
"The players were on their last legs but they have pulled through again. It was a bizarre goal and a bit fortunate, but we thought Mido's shot might have been handled by a defender before the ball ended in the net."
• Sheffield United manager Kevin Blackwell:
"It was a cruel goal to concede, and after all the work-rate and endeavour we put in I thought we might have earned a bit of luck.
"There is nothing Paddy Kenny could have done about it. He is a very good goalkeeper and someone I rate very highly and it is particularly cruel that it should happen to him.
"We can go back home on the bus feeling slightly aggrieved, but the first thing you ask of any group of players is that they give you everything and these lads did that here."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.