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."