Graham Potter's side took an early lead before Amadou Onana leveled the game after 71 minutes.
While the goal itself was perfectly good, it came from a head-scratcher of a decision from the officials to award a corner to Villa after Onana screwed an effort harmlessly wide of the mark.
With no hint of a block or a deflection from a West Ham player, the decision was made to award Unai Emery's side a corner, rather than give Potter's side a goal-kick.
Discussing the incident on ITV, Arsenal legend Ian Wright couldn't believe what he had seen, saying: "The referee and the assistant messed up totally with that corner."
ITV host Mark Pougatch explained: "We should point out that it's not a corner. When you see the replay it's quite clear.
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"Even if there had been VAR, VAR would not get involved in this."
To which an irate Wright continued: "I can't believe the referee, he's standing there.
"He's standing right there and it's gone through a gap of people and he's not seen it.
Look, no one is near it - how can you get that one wrong?
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"West Ham were in this. Yes, they had to defend a bit because Villa got on top but it's such a shame."
From the following corner Onana bundled the ball home to draw the game level at 1-1, before Morgan Rogers gave Aston Villa the lead just five minutes later.
It meant that Emery's side progress to the fourth round of the tournament, while Potter's first match in charge of West Ham ended in defeat.
Potter did, however, make history as the first-ever West Ham boss to make his first managerial appearance for the club in an FA Cup match.
He was also backed by Hammers vice-chairperson Karren Brady, who explained how he was the overwhelming choice to succeed Julen Lopetegui as manager.
West Ham are next in action on Tuesday when they host Fulham in the Premier League.
Meanwhile Aston Villa travel to face Everton on Wednesday night.