Villa missed out on a top-five finish on goal difference to Newcastle and will need to settle for a place in the less lucrative Europa League instead.
The visitors had goalkeeper Emi Martinez sent off before half-time and were furious at the decision to disallow a goal by Morgan Rogers when the score was still level at 0-0.
Referee Thomas Bramall blew for a foul on United goalkeeper Altay Bayindir before Rogers put the ball in the net, meaning VAR could not then review the decision.
To rub salt into Villa's wounds, United then struck through Amad Diallo and Christian Eriksen's penalty to deny them the point they needed.
"The decision is incredible," said Villa captain John McGinn.
"To whistle at that point, it is hard to take. All you need is a point to get into Champions League."
Villa director of football operations Damian Vidagany told reporters the club will complain to the Premier League over the decision to appoint Bramall instead of a more experienced official.