The Toffees registered their first league win of the campaign via a 2-0 success at Birmingham through a Roger Johnson own goal and a late header from Tim Cahill.
It brought an end to City's 13-month unbeaten home record spanning 18 matches.
Moyes said: "Birmingham have a great home record and for us to end it after more than 12 months was really great from our point of view.
"We've now taken four points from away matches at Fulham and Birmingham, two teams I would expect to finish in the top half of the table.
"We played really well, especially in the first half and how we went in 0-0, I don't know, but I've been saying that all season.
"People who have watched us said our performances have not merited where we are in the league.
"The players needed to see the way we were playing was going to get them rewards points-wise - and we got it here.
"Maybe we had a bit of good fortune with the first goal which was deflected but I've been saying maybe we needed something like that.
"We passed the ball well, were fluid and tried to play good football.
"But I am not jumping about because we've won again because I expect us to go on and win a lot of matches."
Moyes admitted that goalscorer Cahill, who nodded in Leighton Baines' cross to seal the win, has still not fully recovered from his recent knee problem.
He said: "There was a doubt about Cahill. He is still not looking 100 per cent.
"He has not trained enough for me to think everything is absolutely right."
The only downside for Moyes the news that midfielder Steven Pienaar could be out of action for a month with a groin problem.
Moyes said: "He hurt his groin on Wednesday in training and it looks as if he might be out for two to four weeks.
"Two weeks would be good, four would not be. It is a real blow because he played so well at Fulham and is an influential player.
"I'm hoping he can heal quickly and at least we have got the international break coming up."
Birmingham manager Alex McLeish admits his side have to start posing more of a threat to opposition defences after drawing a blank at home for the third successive game.
He said: "I have thrown down the gauntlet to all our strikers. Places are up for grabs in the front areas.
"I want to see more in training and reserve matches and the opportunity is there for someone.
"Cameron Jerome is putting himself about up front but we need someone else to be a threat up there."
McLeish was unimpressed with the reaction of some fans who greeted the half-time whistle and final whistle with boos.
He said: "I don't think we deserved it. We need the home support behind us.
"I have been telling them for the last couple of years just how important they are to us."
McLeish added: "We're disappointed to lose the proud home record but now we've got to start again.
"Most teams stay up in this Division by making sure their home form is good."