Danny Ings and Ollie Watkins both put the ball in the back of United's net, but both felt the same frustration as their efforts were ruled out by referee Michael Oliver.
Instead, it was Scott McTominay's eighth minute header which helped the Red Devils to a fourth-round place in the FA Cup.
While Villa manager Steven Gerrard expressed no issue with the actual decisions, he did share his frustrations at the length of time it took for VAR to intervene.
Gerrard was specifically referring to Ings' 51st minute effort which originally stood, but was then ruled out following a near four-minute VAR check.
He told BBC Sport: "How long have you got? The officials took three and a half minutes to settle that one. They have looked at two or three things in it.
"When VAR is there and they make a decision you have to accept it. There is nothing you can do to change it. The easy way out is to blame luck and the officials, we won't do that."
The Villans boss also felt that his side lacked enough ruthlessness to get the job done.
His side produced 11 attempts on goal and even hit the crossbar on what was an impressive outing at Old Trafford.
He added: "It's very tough. We are out of the FA Cup and coming into this job I saw this competition as an opportunity to have a good run in it.
"The lads gave a lot in terms of performance but we need to learn and close the gap from a performance that was nearly good enough. We dominated for large parts but we were not ruthless enough.
"I've seen enough tonight in terms of the performance to know that I can get to where I want to get to, but we need to learn and learn quickly. We need to be more ruthless moving forward, defending crosses we need to be tighter.
"We cannot be having people with free headers in our six-yard box. It is quite clear what we need to iron out and we will do it in the short term."