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Observations of the Milky Way's low-$α$ disk show that at fixed metallicity, [Fe/H], several element abundance, [X/Fe], correlate with age, with unique slopes and small scatters around the age-[X/Fe] relations. In this study, we turn to simulations to explore the age-[X/Fe] relations for the elements C, N, O, Mg, Si, S, and Ca that are traced in a FIRE-2 cosmological zoom-in simulation of a Milky Way-like galaxy, m12i, and understand what physical conditions give rise to the observed age-[X/Fe] trends. We first explore the distributions of mono-age populations in their birth and current locations, [Fe/H], and [X/Fe], and find evidence for inside-out radial growth for stars with ages < 7 Gyr. We then examine the age-[X/Fe] relations across m12i's disk and find that the direction of the trends agree with observations, apart from C, O, and Ca, with remarkably small intrinsic scatters, $σ_{int}$ (0.01-0.04 dex). This $σ_{int}$ measured in the simulations is also metallicity-dependent, with $σ_{int}$ $\approx$ 0.025 dex at [Fe/H]=-0.25 dex versus $σ_{int}$ $\approx$ 0.015 dex at [Fe/H]=0 dex, and a similar metallicity dependence is seen in the GALAH survey for the elements in common. Additionally, we find that $σ_{int}$ is higher in the inner galaxy, where stars are older and formed in less chemically-homogeneous environments. The age-[X/Fe] relations and the small scatter around them indicate that simulations capture similar chemical enrichment variance as observed in the Milky Way, arising from stars sharing similar element abundances at a given birth place and time.