We investigate chemical enrichment in Damped Lyman alpha (DLA) systems in the hierarchical structure formation scenario using a semi-analytic model of galaxy formation. The model developed by Nagashima, Totani, Gouda and Yoshii takes into account various selection effects on high-redshift galaxies and can show fundamental observational properties of galaxies, such as luminosity functions and number-magnitude/redshift relations. DLA systems offer the possibilities of measuring metal abundance more accurately than faint galaxies. For example, recent measurements of zinc abundance can provide good evidence for understanding the processes of metal pollution and star formation in DLA systems because zinc is virtually unaffected by dust depletion. Here we focus on this advantage for observation in order to explore the metallicity evolution in DLA systems at high redshifts. We can consistently show the metallicity evolution for reasonable models which also reproduce fundamental properties of local galaxy population. This result suggests that the chemical evolution of DLA systems can be consistently reconciled with the observational features of typical galaxies. We also investigate other properties of DLA systems (column density distribution and mass density of cold gas), and find that star formation in massive galaxies should be more active than that in low-mass ones. This is consistent with the results by Nagashima et al. and Cole et al. in which the star formation timescale is set by reproducing cold gas mass fraction in local spiral galaxies. Finally we discuss host galaxies associated with DLA systems. We conclude that they primarily consist of sub-L* and/or dwarf galaxies from the observations.