We use cosmological hydrodynamical simulations including star formation and metal enrichment to study the evolution of the chemical properties of galaxy-like objects at high redshift in the range \(0.25<z< 2.35\) in a hierarchical clustering scenario. As the galactic objects are assembled we find that their gaseous components exhibit neutral Hydrogen column densities with abundances and scatter comparable to those observed in damped Lyman-\(\alpha\) systems (DLAs).The unweighted mean of abundance ratios and least square linear regressions through the simulated DLAs yield intrinsic metallicity evolution for the [Zn/H] and [Fe/H], consistent with results obtained from similar analysis of available observations. Our model statistically reproduces the mild evolution detected in the metallicity of the neutral hydrogen content of the Universe, given by mass-weighted means,if observational constraints are considered (as suggested by Boiss\'ee et al. 1998). For the \(\alpha\)-elements in the simulated DLAs, we find neither enhancement nor dependence on metallicity. Our results support the hypotheses that DLAs trace a variety of galactic objects with different formation histories and that both SNI and SNII are contributing to the chemical enrichment of the gas component at least since \(z \approx 2\). This study indicates that DLAs could be understood as the building blocks that merged to form today normal galaxies within a hierarchical clustering scenario.