The Scot described the performance of referee Andre Marriner, and his officials, as "terrible" after Ryan Giggs' own goal and Fabio Borini's penalty left United with a first-leg deficit.
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Giggs diverted into his own net after Marriner awarded a free kick following Jonny Evans' challenge on Steven Fletcher, with Sebastian Larsson's resulting set-piece collected by Wes Brown and fired across the six-yard box where Giggs inadvertently bundled the ball home.
"I thought the referee was going to blow in our direction for the first goal," Moyes continued. "I couldn't believe he gave a free kick. It was a terrible decision.
"We defended a terrible free kick but how the referee has given a free kick for that is way beyond me. I just cannot see how that is a free kick. The centre-half and the centre-forward jumped together and he gave them a free kick."
The Football Association told Goal that Moyes has asked for a non-personal hearing.