We show that the extinction of quasar absorbers increases exponentially with the logarithmic column density of any volatile metal (e.g. zinc), with a characteristic turning point above which the quasars are severely dimmed. We derive the relation between extinction, HI column density ($N_\mathrm{HI}$), metallicity ($Z \equiv N_\mathrm{ZnII}/N_\mathrm{HI}$), and fraction of iron in dust [$f_\mathrm{Fe}(Z)$] in Damped Lyman $\alpha$ (DLA) systems. We use this relation to estimate the effect of dust obscuration on the statistical distributions of $N_\mathrm{HI}$ and $Z$ measured in magnitude-limited surveys of DLAs. In the redshift range where the measurements of zinc column densities have sufficient statistics ($1.8 \leq z \leq 3$) we find that the obscuration bias affects the shapes of the observed distributions. The metallicity distribution is particularly affected by the bias, which hampers the detection of DLA galaxies with near Solar metallicity. Our results perfectly reproduce, without tuning the dust parameters, the DLA observational threshold $\log N(\mathrm{ZnII})\,\mathrm{[cm^{-2}]} \lesssim 13.15$ found by Boissé and collaborators in 1998, in terms of a rapid rise of the obscuration. Our predictions of the effects of the bias on the statistics of DLAs are consistent with observational results obtained from unbiased surveys of radio-selected quasars.