The Islamic Republic of Iran reported its first COVID-19 cases by 19th February 2020, since then it has become one of the most affected countries, with more than 73,000 cases and 4,585 deaths to this date. Spatial modeling could be used to approach an understanding of structural and sociodemographic factors that have impacted COVID-19 spread at a province-level in Iran. Therefore, in the present paper, we developed a spatial statistical approach to describe how COVID-19 cases are spatially distributed and to identify significant spatial clusters of cases and how socioeconomic and climatic features of Iranian provinces might predict the number of cases. The analyses are applied to cumulative cases of the disease from February 19th to March 18th. They correspond to obtaining maps associated with quartiles for rates of COVID-19 cases smoothed through a Bayesian technique and relative risks, the calculation of global (Moran's I) and local indicators of spatial autocorrelation (LISA), both univariate and bivariate, to derive significant clustering, and the fit of a multivariate spatial lag model considering a set of variables potentially affecting the presence of the disease. We identified a cluster of provinces with significantly higher rates of COVID-19 cases around Tehran (p-value< 0.05), indicating that the COVID-19 spread within Iran was spatially correlated. Urbanized, highly connected provinces with older population structures and higher average temperatures were the most susceptible to present a higher number of COVID-19 cases (p-value < 0.05). Interestingly, literacy is a factor that is associated with a decrease in the number of cases (p-value < 0.05), which might be directly related to health literacy and compliance with public health measures. These features indicate that social distancing, protecting older adults, and vulnerable populations, as well as promoting health literacy, might be useful to reduce SARS-CoV-2 spread in Iran. One limitation of our analysis is that the most updated information we found concerning socioeconomic and climatic features is not for 2020, or even for a same year, so that the obtained associations should be interpreted with caution. Our approach could be applied to model COVID-19 outbreaks in other countries with similar characteristics or in case of an upturn in COVID-19 within Iran.