Manager Chris Wilder was already short of numbers in the heart of defence with Chris Basham, John Egan, Jack O'Connell, Jack Rodwell and Jack Robinson all missing the Wednesday evening contest through injury.
The Blades took the lead through David McGoldrick's sixth league goal of the season on the half-hour mark but they were reduced to 10 men just shy of the hour.
Jagielka was booked for fouling Anwar El Ghazi before referee Robert Jones was advised by VAR to look at the incident again, which resulted in him changing his mind and sending Jagielka off for a professional foul.
The victory moves the Blades to 14 points for the season and while relegation is still extremely likely, they won't be going down sharing Derby County's record for fewest points in a top-flight season with 11.
Along with missing so many defenders, the Blades were also without Jayden Bogle and Sander Berge again while Matty Cash and Jack Grealish remained absent for Villa.
Villa had the first big chance of the evening as the ball was played into the Sheffield United area but John McGinn snatched at the shot from six yards as Aaron Ramsdale rushed out to put pressure on the midfielder.
The visitors then had strong appeals for a penalty waved away as Ollie Watkins burst into the penalty area and was challenged by two Blades defenders.
The Villa striker and a few of his teammates called for a foul but nothing was forthcoming from the referee.
McGoldrick got United's noses in front on the half hour. It was the striker's first league goal against a side outside of the 'big six', with all seven of his previous top-flight goals for the Blades coming against Chelsea, Arsenal, Manchester United and Tottenham.
Some remarkable footwork from Bertrand Traore almost hauled the visitors level as he skipped past three defenders to work space for a shot but he put his effort just wide of the post from the centre of the penalty area.
The Blades saw out the rest of the half out before a committed piece of defending from Jagielka two minutes into the second half made sure they stayed in front.
Traoré had a sight of goal after McGinn's flick-on but Jagielka threw his body in front of the ball to divert the danger.
The visitors came close again soon after when Watkins glanced a header towards goal but it came back off the top of the post with Ramsdale beaten.
Jagielka was given his marching orders after a VAR check, as referee Robert Jones deemed his challenge on El Ghazi to be denying a goal-scoring opportunity.
Kean Bryan was on the cover as El Ghazi looked to get in on goal but the referee was happy with his decision to show the red card after checking the VAR monitor.
It is the second time this season the Blades have been reduced to 10 men against Aston Villa when John Egan was sent off for a professional foul in a 1-0 defeat at Villa Park. It was a decision that did not please manager Chris Wilder.
The red card saw Oli Burke replace Rhian Brewster before Ethan Ampadu made an important block to send McGinn's threatening effort behind for a corner.
The visitors were keen to make the most of their man advantage as Ross Barkley and Sanson came on in place of Marvelous Nakamba and Jacob Ramsey.
Villa pressed for an equaliser as Sheffield retreated deeper but Ramsdale was constantly alert to smother the danger, producing a fine save down to his left in stoppage time.
Sheffield United: Ramsdale; Ampadu, Jagielka, Bryan; Baldock, Lundstram, Norwood, Fleck, Stevens; Brewster (Burke 60 (McBurnie 89)), McGoldrick (Osborn 73). Unused substitutes: Foderingham, Sharp, Mousset, Lowe, Osborn, Ndiaye, Maguire.
Aston Villa: Martinez; Elmohamady, Konsa, Mings, Targett; Nakamba (Sanson 68), McGinn, Traore (Davis 81); Ramsey (Barkley 68), El Ghazi, Watkins. Unused substitutes: Heaton, Taylor, Luiz, Engels, Kesler-Hayden.
Referee: R Jones (Merseyside)
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