And, despite Victor Anichebe's sending off and Phil Jagielka's own goal, the visitors held on for the win.
"The players did me proud with their display, especially in the opening 15 or 20 minutes of the match," said Moyes.
"We know the statistics will be about Man City's passes and possession, but the one that mattered was that we got two goals.
"The resilience and competitiveness of the team was excellent.
"Man City played well at times but we defended even better."
Everton had gone seven league games without a win before beating City to move up to 14th in the Premier League table and five points clear of the relegation zone.
"Against City, it was probably the least amount of opportunities we've made [in a match this season]," said Moyes.
"In a way I was disappointed that we didn't get more of the ball, but Man City are an improving team and they showed that."
Moyes was full of praise for his scorers Cahill and Baines, and was particularly impressed by the latter's fine strike on 19 minutes.
"It was a dream second goal, it really was, great movement, great run and a pass into the far corner [of the goal] with Bainesesy's right foot. He should do it more often," said the Toffees boss.
Moyes admitted Anichibe's red card in the 60th minute for two bookings made life difficult, and he disputed the first yellow card shown to the striker.
"I thought the first booking was nothing - he's tried to close it down, he's sort of turned his back, a bit frightened that he was going to get hit with the ball in the face, and his momentum's taken him into the goalkeeper," explained Moyes.
"I think the crowd's reaction got it. I've not seen the second yellow, but it made things really difficult for us.
"We were 2-0 up at the time and doing fine. It was hard to contain them but it was always going to be much harder when we went to 10 men."
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