The Premier League champions went behind thanks to a Ryan Giggs own goal before Nemanja Vidic equalised just after half-time.
However, the hosts were awarded a penalty after Tom Cleverley felled Adam Johnson with Fabio Borini converting from the spot to seal the win.
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"It is difficult we just need to keep going,” he told reporters. “We didn't do a lot wrong. I thought the referee was going to blow in our direction for the first goal. I couldn't believe he gave a free kick. It was a terrible decision.
"We are all just laughing at them at the minute. We get a player booked for doing the same. It looks as though we are having to play them [officials] as well as the opposition at the moment.
"I thought the players, our crowd were unbelievable, the players deserved to get a result. It's a two legged game and I am really looking forward to the second leg."
Moyes also said that reports United have a deal in place to sign Fabio Coentrao from Real Madrid are "not true", before praising the travelling supporters at the Stadium of Light.
“They were brilliant tonight and I would like to thank them all. They supported me and the team,” he added.
"The fans know we’re going through a difficult time and have probably not seen it as bad as this for a long time.
"But I’m determined to put it right and I’m sure we will. I sense that [determination] in the dressing room as well."
The second leg of the Capital One Cup semi-final takes place at Old Trafford on Wednesday 22 January.