Leon Osman and Nikica Jelavic struck as the Toffees responded to their critics after their dismal quarter-final loss to Wigan by beating City for a sixth time in eight meetings.
They had another goal from Kevin Mirallas harshly ruled out for offside and Marouane Fellaini was denied a penalty in the first half.
City could have had their own penalty with the score still 1-0 in the closing minutes but Everton held out despite playing the last half-hour with 10 men after Steven Pienaar was sent off.
Moyes said: "I would have been disappointed if we hadn't responded, and surprised if we hadn't. The players played great. We were playing the champions of England and I thought we gave them a right good run for their money.
"What you saw is what Everton do. I thought we played really well in the first half and I thought in the second half, when we were down to 10 men, we did what Everton did 11 years ago when I took over.
"We were resilient, hard to beat, made sure the opposition didn't score, threw ourselves on the line for the team. We were rubbish last week but we have been good all season and you are watching a really good Everton side.
"It is just to make sure people realise we have not gone away. We are still there. I think people last week thought we had gone away."
Moyes was far from impressed by the sending-off of Pienaar for fouls on Gareth Barry and Javi Garcia.
He said: "I think the sending-off is a terrible decision. The boy rolls about three times and there is hardly any contact. I thought it was the wrong decision."