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The amount of cold, molecular gas in high-redshift galaxies is typically inferred from proxies of molecular hydrogen (H$_2$), such as carbon monoxide (CO) or neutral atomic carbon ([CI]) and molecular gas mass conversion factors. The use of these proxies, however, relies on modeling and observations that have not been directly measured outside the local universe. Here, we use recent samples of high-redshift gamma-ray burst (GRB) and quasar molecular gas absorbers to determine this conversion factor $\alpha_{\rm [CI]}$ from the column density of H$_2$, which gives us the mass per unit column, and the [CI]($J=1$) column density, which provides the luminosity per unit column. This technique allows us to make direct measurements of the relative abundances in high-redshift absorption-selected galaxies. Our sample spans redshifts of z=1.9-3.4 and covers two orders of magnitude in gas-phase metallicity. We find that $\alpha_{\rm [CI]}$ scales linearly with the metallicity: $\log \alpha_{\rm [CI]} = -1.13\times \log(Z/Z_{\odot}) + 1.33$, with a scatter of $\sigma_{\alpha_{\rm [CI]}} = 0.2$ dex. Using a sample of emission-selected galaxies at z~0-5, with both [CI] and CO line detections, we apply the $\alpha_{\rm [CI]}$ conversion to derive independent estimates of the molecular gas mass and $\alpha_{\rm CO}$. We find a remarkable agreement between the molecular gas masses inferred from the absorption-derived $\alpha_{\rm [CI]}$ compared to typical $\alpha_{\rm CO}$-based estimates, and an inferred metallicity evolution of $\alpha_{\rm CO}$ that is consistent with $\alpha_{\rm [CI]}$ and previous estimates from the literature. These results thus support the use of the absorption-derived $\alpha_{\rm [CI]}$ conversion factor for emission-selected star-forming galaxies and demonstrate that both methods probe the same universal properties of molecular gas in the local and high-redshift universe., Comment: 8 pages, 4 figures, 1 table. Abstract abridged. Accepted for publication in ApJL