No slipping off edges with Character Controller

Dear commUnity,

I want to achieve precise and direct 2D platformer controls like in Super Mario World or Braid or other top-quality examples.

Currently, I achieved pretty good controls with my Character Controller. However, it still got a flaw I cannot accept:

My figure is a square and my objects are rectangles. The Character Controller is a pill. So, how do I achieve to NOT slip through the edge of objects. Due to the fact, the CC is rounded, the feet of my character stick inside the object when it is at the edge. At the same time, when dropping off an object it does not continue to run like Super Mario but falling off kind of "automatically".

http://ariest.de/images/edge_problem.png This is an image of the problem I am talking about. Hope this helps you further.

Does anyone know this problem and has someone solved it?

Thank you for reading and answering my question, in advance.