Moyes was infuriated by the decision to award Sunderland a penalty after Tom Cleverley's clumsy tackle on Adam Johnson, with Fabio Borini firing home the resulting spot-kick in the 64th minute of Tuesday night's contest at the Stadium of Light.
And the under-pressure Red Devils boss was equally unimpressed by the free-kick which was given to the home side for Jonny Evans' clumsy aerial clash with Steven Fletcher, with Ryan Giggs eventually putting through his own net after Wes Brown met Seb Larsson's ball into the box.
Moyes, who has overseen United's worst run of form since May 2001 as they suffered a third successive defeat, suggested that decisions from the officials have cost his side dearly.
He told Sky Sports when asked about the penalty: "We had a player booked for the same challenge the other day, for doing the same thing. It's up to the referees, they're making their minds up.
"It looks as if we're having to play them as well as the opposition at the moment.
"It's really terrible, it really is, we're actually beginning to laugh at them.
"We just need to keep going, we didn't do an awful lot wrong. We conceded a terrible first goal from the free-kick, but it was never a free-kick, it was our free-kick.
"I thought the referee was going to blow in our direction. I couldn't believe he gave a free-kick for that Jonny Evans challenge on Fletcher in the air, it was just a terrible decision.
"We've got an opposition to play and sometimes we've got other people to play at the moment."
When asked if he felt under any pressure given United's current woes - they are currently stand 11 points off the pace in the Premier League and are out of the FA Cup - Moyes responded: "None whatsoever."
And Sir Alex Ferguson's successor insists that he is relishing the opportunity to secure a trip to Wembley when the second leg takes place at Old Trafford on January 22.
He added: "This is a two-legged game, and I'm really looking forward to the second leg now.
"Our crowd were unbelievable tonight, the support they gave the team and me was terrific. The players deserved it because they did well enough to get a result.
"We'll do everything that we can to try and get it right. The players put in a strong performance tonight, it just didn't quite come off. It's not far away and it's coming, I can sense it."
Moyes is also adamant Ferguson is a help rather than a hindrance.
Moyes' predecessor was watching from the stands alongside Sir Bobby Charlton and Bryan Robson but the Scot is adamant that the trio are a great help.
Asked if their presence was an added pressure, he said: "Not at all.
"Alex is a great sounding-board for me. He is there to help, so is Sir Bobby. He is terrific. Bryan Robson is at the training ground regularly, I speak to him regularly.
"I find the three of them terrific."