The answer is that you have overlooked two rather important features of evolution
- Evolution does not mean survival of the fittest. It means survival of the sexiest.
- Mutation is heriditary
Survival of the sexiest means that organisms that are most successful in getting a mate, not necessarily the fittest, or the strongest or the fastest are the ones that propogate their genes to the next generation. A classic example of this is the peacock. The feathers of the male peacock is a nuisance to carry around and make the peacock an easy meal for many of the big cats. But evolution has favoured the long tail because the peahens are most attracted to the peacock with the longest feathers. If it was the feathers for peacock, it was brain (intelligence) for humans.
Now, the genetic material of living organisms is very long, and most of it does not serve any apparent purpose. Most of the mutations occur in this dysfunctional part of the DNA and hence its effects aren't observed. Mutation isn't really as rare as people think it is.
Among the mutations that occur in the active part of the DNA, most are harmful in various degrees. These are the mutations that lead to cancer, third nipple and the sixth finger,for example. And obviously these freaks don't get to mate. Since these mutations don't pass on to the next generation. On the other hand, those mutations that are favourable, despite the fact that they occur very rarely, are successful in attracting a mate, and hence gets passed on to the next generation, and from there it is an exponential growth in numbers.
A random process can indeed lead to evolution.
To be continued...