A goal from Carlos Tevez - his 10th of the season - in the fourth minute was enough to separate the side.
While Mancini was pleased they kept the pressure on the leaders he despaired at the number of chances that went begging. Mario Balotelli and David Silva both saw efforts come back off the woodwork at Eastlands.
Mancini said: "You can say we had some bad luck but we must improve. Sometimes we are too selfish. We had maybe 15 chances to score. When you have a lot of chances you think the game is easy.
"You shoot when it is better to pass the ball. We must improve this situation. It means you can draw when you should have won by three or four goals.
"It was the same at Stoke where we deserved to win but we drew. When we have a chance we must score. If I had played I would have scored two or three goals."