The earth/moon system is special:
If you calculate the gravitational forces on the moon, the largest contribution comes from the sun. In the reference frame of the sun, the path of the moon is always bent towards the sun. To see that the moon orbits the earth, you have to look at its position relative to earth, and ignore the common orbit of earth+moon around the sun.
I might be off-base in mentioning this, but it appears to me that there's another factor which is unusual in orbital situations. The moon is so huge in comparison to other satellites as to mass relative to the host planet, it's own gravity has a pretty strong influence as well (thinking of tides here).