I am not aware of a similar myth in other world mythologies, but I do know that three major religions all have explanations as to how evil entered the world. So, perhaps you could compare the myth of Pandora's Box to the idea of evil entering the world. If you do some research into Christianity and Judaisim, for example, you will find that evil entered the world in the Garden of Eden, when Satan, in the form of a snake, tempted Eve to take a bite of the fruit of the tree of knowledge. God specifically told Adam and Eve not to eat from that tree because they would die. They disobeyed God, and as a result, sin entered the world. Since Christianity and Judaisim view sin as rebellion against God, all sin stems from this first act of disobedience. Satan lied to Eve, telling her, "Surely you will not die!" What God meant was that if they did not eat, they would live forever, but if they did eat, they would no longer have eternal life. As punishment for disobedience, Adam and Eve were expelled from the garden and no longer had eternal life. So, Adam and Eve are the Pandoras.
You could also research the explanation for evil entering the world in other religions. I know there are some interesting stories in the Native American legends as well.