Ezri Konsa's second-half header proved decisive as Aston Villa overcame a resilient performance by Sheffield United, who played for 78 minutes with 10 men after the early dismissal of John Egan.
Konsa headed in fellow defender Tyrone Mings' flick-on in the 63rd minute after Matt Targett whipped a corner in from the left.
Villa put United under pressure for long spells after visiting captain Egan was sent off for denying a clear goalscoring opportunity in a tussle with Ollie Watkins in the 12th minute.
But the Blades missed a golden opportunity to score the opener after half an hour when John Lundstram was denied from the penalty spot by Villa debutant Emiliano Martinez, after a lengthy VAR review showed Targett had clipped Chris Basham in the box.