The Black Cats produced a good performance to take the tie back to the Stadium of Light and Moyes said Martin O'Neill's side were now the favourites.
"I think the pressure does go back on them. I felt it a little bit when we played at home," said the Scot. "We tried to do the job and win here but it was a good result for Sunderland to get us back to the Stadium of Light."
He added: "They will see a good part of the job has been done and they have probably got themselves in the favourites position because they have been very good at home under Martin, so we know it will be a tough game."
Tantalisingly for Everton, a semi-final against near-neighbours Liverpool awaits the winners but Moyes is keen for the focus to be on the job in hand.
The Toffees have not been to Wembley since they lost the 2009 final to Chelsea, which was the closest the club have come to ending their 17-year trophy drought.
"It would be great. It is something I would really look forward to," said Moyes when asked for his thoughts on returning to the national stadium. "I really enjoyed the last time - except for the result - but it would be terrific if we could go and do it again.
"But I am not thinking or talking that way because I would rather try to play in this game and then see where we go from there. It is too soon to ask me a question about getting back to Wembley at the moment."
Moyes' greater concern is fatigue for his players with the cup tie their fifth match in 17 days. Part of the congestion was caused by the re-scheduling of the Merseyside derby after Liverpool won the Carling Cup, but the Everton boss does not believe the Premier League have helped their cause.
"It will affect the squad but you have to take the games as they come," he said of their recent run of fixtures. "You have to get on with it but, as I've said a couple of times, I don't think it has been particularly fair on us. The Premier League have been hard on Everton this year, that is for sure."
Source: PA
Source: PA