The prime suspect for injecting Biblical references into cosmology would be Georges Lemaître, a cosmologist and... a Catholic priest. His "Cosmic Egg" and "Primeval Atom" lost to Hoyle's sarcastic "Big Bang" however, and it seems he passed up on introducing "dust". His 1927 Ph.D. thesis The Gravitational Field in a Fluid became the basis for the now famous 1927 paper Homogeneous Universe of Constant Mass and Growing Radius Accounting for the Radial Velocity of Extragalactic Nebulae, written in French, and self-translated into English in 1931 with minor edits. Lemaître talks about nebulae and introduces the usual assumption of negligible pressure, but does not call his fluid "dust" even descriptively.
So it seems the 'priority' goes to Tolman's Effect of Inhomogeneity on Cosmological Models (1934), who writes "the fluid in the models was taken as dust exerting negligible pressure", and repeats "dust" 16 times on 8 pages. In the assumptions on the particular form of stress-energy tensor he follows Lemaître:"we shall confine our attention to very simple models composed of dust particles (nebulae) which exert negligible pressure... This will permit us to employ expressions for the line element and its consequences which are equivalent to those recently developed by Lemaître for investigating the formation of nebulae". Some authors define "dust" more broadly, Sachs and Wu in General Relativity for Mathematicians (3.13) simply give a kind of divergence free condition on stress energy tensor, and note that pressureless perfect fluid satisfies it. The prototypical example seems to be nebulae, clouds of particles interacting only gravitationally, literally dust.
Although the association with stress energy tensors is clearly specific to relativity more loosely "nebulae" and "dust" were already used by Newtonian astronomers, and there seems to be continuity with Tolman's usage. The reference to nebulae reminds the nebular hypothesis, according to which a rotating dust cloud formed the solar system. It goes back to Kant (1755), and in a more detailed form to Laplace (1796), and was revived by Chamberlin and Moulton in 1901. Both van Stockum's and Gödel's early "dust solutions" to the Einstein equations involve rotating dust. The nebular hypothesis is (metaphorically) in line with the Biblical origin story, but nebular dust is also literally analogous to the ordinary dust, so we can only speculate what contributed to the spread of usage. As an alternative reference, Democritus envisioned a universe filled with "fine dust which is divided in the breeze and becomes visible in the sun's beam". Lucretius expressed it poetically in De Rerum Natura, well familiar to people with classical education.
There are other examples of classical usage. The paradox that in an infinite universe uniformly filled with stars the night sky would appear to be bright, stated most eloquently by Olbers in 1826 and named after him, was already pointed out by Digges back in 1576. Kepler took it as an argument for finite universe, but Halley in 1705, Cheseaux in 1744 and Olbers himself suggested interstellar dust obscuring distant stars as a resolution. In 1831 Herschel pointed out however that the dust would simply be heated up until it glowed as brightly as the stars (further reflection shows that the sky should in fact be infinitely bright). Finally, zodiacal light is explained by an interplanetary dust cloud, which attracted much attention at the turn of the centuries, because both Newcomb (1895) and Seeliger (1906) tried to use it to explain the anomalous precession of Mercury, before Einstein gave the general relativity explanation in 1915.