All eyes were on Villa's new record signing Ollie Watkins but it was defender Konsa, a former team mate of Watkins' at Brentford, and debutant goalkeeper Emiliano Martinez who earned their side three points.
Sheffield United had defender John Egan sent off after 12 minutes for pulling back Watkins but wasted a great chance to take the lead before half-time when John Lundstrum's penalty was saved by Martinez, who signed from Arsenal last week.
Even with an extra man Villa struggled to create chances against a well-organised United side, but Konsa's 63rd minute header ensured a positive start for Dean Smith's side who last season escaped relegation by the skin of their teeth.
It has been a disappointing start to the campaign for Chris Wilder's United side who have lost two from two without scoring.
Smith said Egan's sending off had forced his Villa side to throw their game plan out of the window.
"We thought United would come and press and we wanted to suck them in and then play in the spaces behind," he said. "But that high press stopped as soon as Egan was sent off."
The last time the two sides met -- the first Premier League match after the resumption of last season in June -- the game was mired in controversy when United were denied a clear goal because of a failure of goal-line technology.
And they could feel a little hard done by again as Egan tugged at the shirt of Watkins, who was trying to latch on to a long pass by Tyrone Mings. Referee Graham Scott was a long way from the incident but had no hesitation in showing a red card.
United looked comfortable but their resistance was broken when Targett swung in a corner, Mings glanced it on and Konsa's header, lacking any power, crept past keeper Aaron Ramsdale.
Villa captain Jack Grealish curled a right-foot effort just wide and Watkins, who scored 26 goals in all competitions for second-tier Brentford last season before his £27 million move, also went close with a similar effort.
Villa's scheduled opener at Manchester City on Sept. 12 was postponed so that Pep Guardiola's side could have an extra week to recover from their Champions League campaign last term.