In the Epic of Gilgamesh the protagonist defeats, with the aid of Enkidu, two major beastlike characters.
Humbaba is the guardian of the Forest of Cedars. Gilgamesh and Enkidu battle him to fell the trees and collect their timber as well as to achieve glory for their martial prowess.
The Bull of Heaven is sent by Ishtar for revenge when she is angered by Gilgamesh's rejection of the role as her consort. The Bull represents drought and famine, and in defeating it Gilgamesh proves his worth as a king.
There really is no equivalent beast in the Garden of Eden story; Satan in the form of a serpent is really a tempter that reveals Adam's own moral weakness rather than a traditional beast to be battled. The best Biblical parallel would be the story of David and Goliath.
There are no clear parallels in Genesis to the battles in Gilgamesh. However, there are some minor similarities between the Bull of Heaven episode and the story of Joseph. In each the hero, who is described as well built and handsome, rejects the advances of a woman (Ishtar and Potiphar's wife). Ishtar's screaming in one version of the story (a Sumerian Gilgamesh poem), parallels the references to the woman's screaming in the Genesis story. In both, the hero is accused before a third party and a punishment is decided upon. Then, interestingly, each story contains a reference to seven years of famine. In Gilgamesh, Anu says to Ishtar:
If you demand the Bull of Heaven from me,
there will be seven years of empty husks for the land of Uruk.
This parallels Joseph's interpretation of Pharaoh's dream. Relief is provided in both stories by the storage of grain:
Have you collected grain for the people!
Have you made grasses grow for the animals?"
Ishtar addressed Anu, her father, saying:
"I have heaped grain in the granaries for the people,
I made grasses grow for the animals,
in order that they might eat in the seven years of empty husks.
In Gilgamesh, the story segues into a dream in which one of the heroes in sentenced to death and the other goes unpunished. This recalls the dreams of the baker and cup bearer, in which one is condemned and the other elevated.