City went ahead through Yaya Toure and then squandered a number of chances before Everton hit back through former Blues defender Sylvain Distin and Leon Osman.
Moyes said: "They bossed the game and we never laid a glove on them. We were fortunate to go in at half-time 1-0 down.
"We changed the system round a little bit at half-time and I don't know if it was the system but we got closer to people, looked more attacking and were a bigger threat.
"We set out to try to contain Man City but it is very difficult to contain David Silva. I think he is as good as anyone in the Premier League at the moment."
Source: PA
Source: PA