Evil can only exist consequently. Consequent of something God did? Well, yes, but not directly. God did not breath evil into existence, did not cogitate it and then induce it. Evil does not arise from the heart of God, though evil has most certainly arisen.
Evil is not a creation of God, but a creation of creations. Evil bursts into existence when a creation created with the capacity to itself freely create, creates against the will of God. That creation can be an attitude, an action, an aim, or whatever, but to be evil it must be at odds with God--that's what makes it evil. God's response to such is corrective and terminal.
Given God's omni characteristics, if evil were allowed to continue, it would be as if he harbored evil within himself. That would be schizophrenic and impossible, for a house divided against itself cannot stand. Because God is everywhere always, evil cannot be. Therefore, God has circumscribed evil within time for a time until he deals with it eternally.
From the perspective outside of time, evil has not even been a hiccup. For us in time it seems as if evil has been around forever (and always will be), but that is only an illusion we experience from our perspective within time. Outside of time, this realm of time is not so much as even a flash. God who is always the same is there "already" at the beginning and end, whereas evil is not.
Evil is not self-existent, not eternal, and not therefore truly something. Although here and now it is real and has far-reaching consequences, in the eternal scheme of things it is as nothing. Evil is consequent (and therefore dependent) upon creatures taking free actions which are in opposition of God. Where there is no will in opposition to God, there is no evil.