(1) Background: Research highlights the positive effects of early intensive intervention with parent and school involvement for preschool children with Autism Spectrum Disorder (ASD) on general developmental outcomes and social skills in randomized controlled trials. However, given the inter-individual variability in the response to treatment, it is necessary to investigate intervention effects in terms of mediators and moderators in order to explain variability and to highlight mechanisms of change. (2) Methods: 25 children in the experimental group were exposed to early intensive intervention and 14 children in the control group were subjected to "as usual" intervention. The initial assessment was obtained at the time of diagnosis (T1) and the follow-up assessment was conducted after 15 months of intervention (T2) in both groups. (3) Results: Participants in the experimental group achieved more prominent gains in both cognitive and socio-interactive skills. The role of specific factors able to predict general quotient and language quotient after intervention were investigated, pointing out the contribution of personal-social and performance abilities. (4) Conclusions: The findings support the importance of parental involvement in targeting ASD core symptoms. Further, results informed our understanding of early predictors in order to identify specific elements to be targeted in the individualized intervention design.