Individuals with alcohol use disorder (AUD) are known to manifest a variety of neurocognitive impairments that can be attributed to alterations in specific brain networks. The current study aims to identify specific features of brain connectivity, neuropsychological performance, and impulsivity traits that can classify adult males with AUD (n = 30) from healthy controls (CTL, n = 30) using the Random Forest (RF) classification method. The predictor variables were: (i) fMRI-based within-network functional connectivity (FC) of the Default Mode Network (DMN), (ii) neuropsychological scores from the Tower of London Test (TOLT), and the Visual Span Test (VST), and (iii) impulsivity factors from the Barratt Impulsiveness Scale (BIS). The RF model, with a classification accuracy of 76.67%, identified fourteen DMN connections, two neuropsychological variables (memory span and total correct scores of the forward condition of the VST), and all impulsivity factors as significantly important for classifying participants into either the AUD or CTL group. Specifically, the AUD group manifested hyperconnectivity across the bilateral anterior cingulate cortex and the prefrontal cortex as well as between the bilateral posterior cingulate cortex and the left inferior parietal lobule, while showing hypoconnectivity in long-range anterior–posterior and interhemispheric long-range connections. Individuals with AUD also showed poorer memory performance and increased impulsivity compared to CTL individuals. Furthermore, there were significant associations among FC, impulsivity, neuropsychological performance, and AUD status. These results confirm the previous findings that alterations in specific brain networks coupled with poor neuropsychological functioning and heightened impulsivity may characterize individuals with AUD, who can be efficiently identified using classification algorithms such as Random Forest.