Interpreting abundances of damped Ly-alpha absorbers (DLAs) from absorption-line spectroscopy has typically been a challenge because of the presence of dust. Nevertheless, because DLAs trace distant gas-rich galaxies regardless of their luminosity, they provide an attractive way of measuring the evolution of the metallicity of the neutral gas with cosmic time. This has been done extensively so far, but typically not taking proper dust corrections into account. The aims of this paper are to: (i) provide a simplified way of calculating dust corrections, based on a single observed [X/Fe], (ii) assess the importance of dust corrections for DLA metallicities and their evolution, and (iii) investigate the cosmic evolution of iron for a large DLA sample. We have derived dust corrections based on the observed [Zn/Fe], [Si/Fe], or [S/Fe], and confirmed their robustness. We present dust-corrected metallicities in a scale of [Fe/H](tot) for 236 DLAs over a broad range of z, and assess the extent of dust corrections for different metals at different metallicities. Dust corrections in DLAs are important even for Zn (typically of 0.1-0.2, and up to 0.5 dex), which is often neglected. Finally, we study the evolution of the dust-corrected metallicity with z. The DLA metallicities decrease with redshift, by a factor of 50-100 from today to similar to 12.6 billion years ago (z = 5). When including dust corrections, the average DLA metallicities are 0.4-0.5 dex higher than without corrections. The upper envelope of the relation between metallicity and z reaches solar metallicity at z less than or similar to 0.5, although some systems can have solar metallicity already out to z similar to 3.