Goals in victory over Aston Villa from man-of-the-match Mario Balotelli, Adam Johnson, Vincent Kompany and James Milner took City to the top of the table after Liverpool and Manchester United drew 1-1 at Anfield.
Mancini said: "We've only played eight games, there are another 30, it's a very long way. We should be happy to go to Old Trafford on top but we should understand that the position can change every week.
"It's important to stay there or very close now. We will fight until the end. We have started very well and won a lot of games, but we need to do more."