To answer the question:
Of course every mythology feels the need to explain the good/evil difference. As I see, most mythologies in the world explain it by bipolarity (e.g. the North and South in a magnet cannot be set apart, so the good and evil are both necessary parts of every individual -- of course not so balanced as the simpler magnetic analogy, but they can't be isolated as well). Monotheisms explain it by dualism (like Platonic reality/ideality, good and evil are isolated things, so one person may be 100% good, and another 100% evil. That's the basis of maniqueism, and also a root for the melodrama that stains most of our cultural production, instead of the richer Greek tradition of tragedy, as illustrated by Sophocles' Oedipus). You can read more about this distinction in Marc Halévy's Lecture du Tao: une sagesse qui nous attend... (2012).
Now some more detail, about the question itself and about virmaior's answer:
I doubt that "shan" alone may represent our word "good", as well as "e" alone our word "evil". That's because hao 好, liang 良, jia 佳, guai 乖, mei 美, jia 嘉, as well as lianghao 良好, youliang 优良, nashou 拿手, bulai 不赖... all mean "good". So the translation here is far from direct or simple. If "to know" may be translated by more than 10 different characters, then we're probably "out of words". Both Confucius' Analects and Orwell's 1984 have taught us about the importance of the right words, but the West doesn't seem to be listening (or reading).
Also, this methodology brings its own problems: guai 乖 is also "perverse", "contrary to reason", "abnormal". A similar problem in translation is chang 常, which may be translated as "common", "usual" and at the same time "eternal", "unchanging". This simple character interfered with centuries of Western translations of the Chinese texts (most translators, saturated with monotheism, choose the latter meaning of chang, although the former is more significant in the light of both the remaining text and the findings in animal behavior or physics).
That being said, primitive Daoism apparently agrees with Mengzi's vision in virmaior's 3rd paragraph. The Dao De Jing (57) says "I do nothing and the people are reformed of themselves. I love quietude and the people are righteous of themselves. I deal in no business and the people grow rich by themselves. I have no desires and the people are simple and honest by themselves." [Lin Yutang 1955 http://web.archive.org/web/20110514203735/http://home.pages.at/onkellotus/TTK/English_Yutang_TTK.html#Kap57] That's a great definition of "good" to me, though the character shan 善 is absent.
In a modern, scientific world-view, biology says that any social species will know how to tell good from evil. That's illustrated in Jane Goodall's book "Through a Window", when the group of Gombe chimpanzees isolates an old female who kills an infant.
This Daoist point of view is also exemplified in the movie K-PAX (2001), where Kevin Spacey's character replies to his restless, frustrated doctor, who is asking about his "other-worldly society", which is described as having no rules or government. The doctor asks "How do you know right from wrong?", to which the "alien" answers the obvious: "Every being in the Universe knows right from wrong, Mark." That's a great movie about Daoist thinking. Remembering that the society described by the "alien" is strikingly similar to those existent all over America, which were unmercifully destroyed by so-called Christians "devoted to good".
So, to conclude: yes, Eastern philosophies are well aware of the good/evil problem, but no, they don't posit this two concepts as "dueling", but are able to recognize their "complementarity", opposed to those who, trying to bring us "100% good" actually brought some of us "100% evil"...