Villa goalkeeper Orjan Nyland appeared to have carried a free-kick over the line but referee Michael Oliver did not receive a signal to award the goal.
The company - who operate the Premier League's goalline technology system - said: "Hawk-Eye unreservedly apologises to the Premier League, Sheffield United and everyone affected by this incident."
The firm added that it was the first time that a mistake had occurred in over 9,000 matches using the device.
It also explained that all seven cameras were obstructed.
United boss Chris Wilder blasted the decision, which cost the Blades a win as the match at Villa Park ended up goalless.
The 52-year-old manager said: "I think the goalkeeper was in the Holte End when he caught it.
"Seven cameras haven't picked it up - the most technical league in the world - everything we see at every angle and it hasn't seen a goal."
Wilder also bemoaned the fact that VAR wasn't used for the incident, citing a decision that went against them when they played Tottenham earlier this season.
He added: "We went to Tottenham last year, John Lundstram's big toe was offside and I was hanging around in the pouring rain for 10 minutes waiting for a decision. I don't understand why I can't wait 10 minutes in Birmingham."
Villa boss Dean Smith admitted that his side had had a lucky escape with the decision.
He said: "We got away with one certainly with the technology not working."