Supply chains are complex networks that receive assiduous attention in the literature. Like any complex network, a supply chain is subject to a wide variety of risks that can result in significant economic losses and negative impacts in terms of image and prestige for companies. In circumstances of aggressive competition among companies, effective management of supply chain risks (SCRs) is crucial, and is currently a very active field of research. Failure Mode, Effects and Criticality Analysis (FMECA) has been recently extended to SCR identification and prioritization, aiming at reducing potential losses caused by lack of risk control. This article has a twofold objective. First, SCR assessment is investigated, and a comprehensive list of specific risks related to the automotive industry is compiled to extend the set of most commonly considered risks. Second, an alternative way of calculating the Risk Priority Number (RPN) is proposed within the FMECA framework by means of an integrated Multi-Criteria Decision-Making (MCDM) approach. We give a new calculation procedure by making use of the Analytic Hierarchy Process (AHP) to derive factors weights, and then the fuzzy Decision-Making Trial and Evaluation Laboratory (DEMATEL) to evaluate the new factor of “dependence” among the risks. The developed joint analysis constitutes a risk analysis support tool for criticality in systems engineering. The approach also deals with uncertainty and vagueness associated with input data through the use of fuzzy numbers. The results obtained from a relevant case study in the automotive industry showcase the effectiveness of this approach, which brings important value to those companies: When planning interventions of prevention/mitigation, primary importance should be given to (1) supply chain disruptions due to natural disasters; (2) manufacturing facilities, human resources, policies and breakdown processes; and (3) inefficient transport.