Manuel Pellegrini's men were 2-0 up at half-time but the hosts levelled matters in the second half and City needed further strikes from Jesus Navas and Milner to secure the win.
The result leaves them a point behind leaders Liverpool, whom they face on Boxing Day, and the Englishman wants his side to be more conservative.
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"It would have made Christmas dinner taste worse for the fans, so it was good to get the win before two games in quick succession. We now have the momentum to go into those games and hopefully get six points."
The 27-year-old also urged his side to improve defensively after keeping just one clean sheet in their last eight matches in all competitions.
"You can't get anywhere in football without scoring goals but, as a side that's looking to improve, we can improve at the other end, as a team," he added. "Hopefully we can get a few more clean sheets, which will be more important as the season goes on.
"We need to work on the mistakes we are making throughout the team. When we are conceding goals it's not the defence and the goalkeeper [to blame], it's the whole team.