Are there any natural animal enemy to cockroach to eliminate them all? ?

Question

So we do not need to use other side effect things to eliminate them?

Answer 1

I wish.  Wouldn't it be great if cats ate cockroaches?

Answer 2

No predator can hunt the prey species to extinction, or it would die out too.  

One of the best methods of killing cockroaches is to use boric acid.  It is sold as a bug powder, and is harmful only to animals with exoskeletons (e.g. insects and spiders).  It damages the exoskeleton, causing them to dry up.  It is safe around children and pets like birds, mammals, and reptiles.  

It comes in a puff bottle and you puff a thin layer anywhere the cockroaches may pass by.  They will avoid a pile of it, so a very thin puff is the best.  You can also mix it with their favorite food, e.g. mashed peas.  They will not die instantly, but they will die soon.

I use it for ants and fleas.

It may be a job to exterminate them entirely, but it can be done, unless there is a source of them coming in from elsewhere.