I have heard that The Epic of Gilgamesh may predate the stories of Noah told in Abrahamic works. Is there any evidence to support which story came first?
The "standard" version of the Epic of Gilgamesh, the one edited by Sîn-lēqi-unninni, is dated to somewhere between 1300 BCE and 1000 BCE. Although the epic is far earlier, this is the version that includes Utanapishtim's story of the flood.
The dating of Genesis, however, as many matters that may have some religious significance is not entirely concluded. Depending on who you ask, you will receive answers ranging from 1400 BCE to the 5th century BCE, with the earlier limit of the range stemming from traditional views rather than scientific methods (e.g. the disputed concept of Mosaic authorship). If you accept the more conservative estimations of about 700 BCE, then the flood myth in Gilgamesh is clearly earlier.
If on the other hand, you wish to allow for the wildly exaggerated traditional datings, then I think you should also permit for a small switch of focus from Gilgamesh to the Akkadian Epic of Atra-Hasis. Atra-Hasis also includes a flood myth, and it's widely accepted1 that the flood myth in Tablet XI of the aforementioned "standard" version of Gilgamesh derives from it.
Furthermore, the flood myth in Gilgamesh is not directly comparable to the biblical narrative. In Gilgamesh, the tale of the flood is secondary to the eponymous hero's quest and is severely restricted by its first person narrative; Utanapishtim can only tell us what he personally experienced. One of the more important things we never learn, for example, is the reason for the flood, as, presumably, Utanapishtim was not aware of it.
Atra-Hasis, on the other hand, provides us with the history of the world before the creation of man, and a fuller story of the deluge, including an explanation of its cause. It is, therefore, a lot more comparable to Genesis than Gilgamesh, and this I believe is why we should focus on it instead. Following this path will lead us to an even clearer answer to your question as the earliest known record of Atra-Hasis can be dated to around 1646–1626 BCE (during the reign of Ammi-Saduqa), by its colophon. Thus, Atra-Hasis is earlier even if we accept Moses wrote Genesis.
Lastly, I would be remiss not to mention that the other Mesopotamian flood story, the Sumerian tale of Ziusudra, is also older than Genesis. The story is told in a single tablet, that has been dated to around 1600 BCE.
1 "In the intervening years, it is true, it had become clear that what Smith had discovered, the eleventh tablet of the Gilgamesh Epic, was derivative" - Moran, William L. “Atrahasis: The Babylonian Story of the Flood.” Biblica, vol. 52, no. 1, 1971, pp. 51–61. www.jstor.org/stable/42609705.
It's unknown, but it appears that although The Epic of Gilgamesh was written first, the Pentateuch (part of the Bible containing the creation stories) has the first description of the flood story.
From the Idaho University page on The Epic of Gilgamesh:
The oldest existing versions of this poem date to c 2000 BC, in Sumerian cuneiform. The more complete versions date to c. 700 BC, in the Akkadian language. The standard, first "complete" version, which includes the flood myth, is dated to c. 1300-1000 BC.
TL;DR: Oldest version of the poem is from around 2000 BC, the first version to depict the flood is around 1300-1000 BC.
In the book The City of God by Saint Augustine (best reference I can find), Moses is told to have been born in 1400 BC and died in 1201 BC.
Augustine also pens Moses as the author of the whole Pentateuch, so the flood story was written by him at some point during his lifetime.
So the Biblical flood story was written from around 1430-1201 BC, whereas the first version of The Epic of Gilgamesh to contain the flood story was written from 1300-1000 BC.
So it's possible either way, but it's most likely they were written somewhat concurrently.
Do note that it's unknown for sure whether Moses wrote the Pentateuch, but since the end of Genesis/beginning of Exodus (the first 2 books) discuss Moses, it's necessarily at this time period or after. (IIRC there is a 50 year gap between the books however)
Walther Sallaberger's book Das Gilgamesch-Epos. Mythos, Werk und Tradition (C. H. Beck, 2008) discusses the flood story mainly in the context of the Standard Babylonian version of the Gilgamesh Epic. This is the version whose redaction is attributed to Sîn-lēqi-unninni and which Sallaberger dates to the 11th century BC. There is no evidence that the flood story was part of earlier Gilgamesh stories, such as the five or six Sumerian stories (which do not constitute a continuous narrative), the Old Babylonian version (which is very fragmentary but appears to have been an integrated epic.), the Hittite version (which was shorter than the Standard Babylonian version and in prose) or in other older versions.
However, the flood story on Tablet XI of the Standard Babylonian version of the epic did not come out of the blue. The story on Tablet XI is based on the Atrahasis story, which existed in several versions. Benjamin R. Foster's anthology Before the Muses: An Anthology of Akkadian Literature (CDL Press, 2005) provides several versions (translations of the fragments that have been discovered so far): an Old Babylonian version, a Middle Babylonian version (a dozen incomplete lines from a tablet found in Nippur), a Late Babylonian version and a Late Assyrian version. The Classical or Old Babylonian period lasted approximately from 1850 to 1500 BC.
That the Standard Babylonian version used the Atrahasis story is obvious from verbal parallels, including an occurrence of the name "Atrahasis" instead of Utanapishtim in Ea's speech in Tablet XI.
If we accept the Standard Babylonian version of the Gilgamesh Epic as the earliest Gilgamesh story that contains the flood, it is clear that this story is older than the story in the Book of Genesis, which probably dates from the 6th and 5th centuries BC.
Note: user8's answer appears to misinterpret the phrase "Standard Babylonian version" as "standard version". However, "Standard" is capitalised because it refers to Babylonian (as in "Standard Chinese") and not to "version". Standard Babylonian refers to a specific literary style used during the Mature and Late periods of Akkadian literature. This style can be distinguished from the older Hymnic-Epic style. (See Benjamin R. Foster: Before the Muses: An Anthology of Akkadian Literature, CDL Press, 2005, pages 3-4.) The Standard Babylonian version may be called the “standard version” because that is the version that remained (relatively) stable until the last extant manuscript of the epic.