Accurate solar irradiance forecasts are now key to successfully integrate the (variable) production from large solar energy systems into the electricity grid. This paper describes a wrapper forecasting methodology for irradiance time series that combines mutual information and an Extreme Learning Machine (ELM), with application to short forecast horizons between 5-min and 3-hour ahead. The method is referred to as Wrapper Mutual Information Methodology (WMIM). To evaluate the proposed approach, its performance is compared to that of three dimensionality reduction scenarios: full space (latest 50 variables), partial space (latest 5 variables), and the usual Principal Component Analysis (PCA). Based on measured irradiance data from two arid sites (Madina and Tamanrasset), the present results reveal that the reduction of the historical input space increases the forecasting performance of global solar radiation. In the case of Madina and forecast horizons from 5-min to 30-min ahead, the WMIM forecasts have a better coefficient of determination (R2 between 0.927 and 0.967) than those using the next best performing strategy, PCA (R2 between 0.921 and 0.959). The Mean Absolute Percentage Error (MAP) is also better for WMIM [7.4–10.77] than for PCA [8.4–11.55]. In the case of Tamanrasset and forecasting horizons from 1-hour to 3-hours ahead, the WMIM forecasts have an R2 between 0.883 and 0.957, slightly better than the next best performing strategy (PCA) (R2 between 0.873 and 0.910). The Normalized Mean Squared Error (NMSE) is similarly better for WMIM [0.048–0.128] than for PCA [0.105–0.130]. It is also found that the ELM technique is considerably more computationally efficient than the more conventional Multi Layer Perceptron (MLP). It is concluded that the proposed mutual information-based variable selection method has the potential to outperform various other proposed techniques in terms of prediction performance.