We can not do this by comparing the dates of the oldest extant manuscripts/tablets of Genesis and Enuma Elish, because all we have in both cases are copies of copies. However, we can examine the Enuma Elish story to establish what period is referred to.
Leonard W. King (Enuma Elish, pages LXXII-LXXX) says of the actual tablets inscribed with portions of the creation series, we posses none that dates before the seventh century BCE. The tablets of this date were made for the library of Ashur-bani-pal in Nineveh, but it is obvious that the poem was not composed in Assyria at this time. The legends in the form that we have them were not intended to glorify Ashur, the national god of Assyria, but Marduk the god of Babylon, and it is clear that the scribes of Ashur-bani-pal merely made copies for their master of older tablets of Babylonian origin. He says evidence of the Creation legends at least as early as the ninth century BCE may be deduced from representations of the fight between Marduk and the dragon Tiamat, which was sculptured on two limestone slabs in the temple of Ninib in Nimrud. Further evidence that appears to take us back at least to the seventeenth century BCE is found in a text of the Babylonian king Agum. King then describes some much earlier texts similar to the Enuma Elish legends, and believes the legends behind Enuma Elish may date back to the first half of the third millennium or even earlier, although this takes us into the realm of speculation.
The question presupposes that parallels exist between the Enuma Elish and Genesis, so I do not propose to go over that ground. Biblical tradition says that Moses wrote Genesis around 1400 BCE, which date is after the time of the legends on which the Enuma Elish is based. Scholars date the actual authorship of Genesis to the first millennium BCE, with the first creation story (Genesis 1:1-2:4a) attributed to the Priestly Source, writing during or shortly after the Babylonian Exile. The evidence that the Enuma Elish already existed before the Genesis creation stories is quite strong.