Background: Green Public Procurement (GPP) is a sector that has been growing in recent years through policies encouraged by the European Union. In the Greek public sector, the respective National Action Plan (NAP), which sets specific targets for GPP, has very recently come into force. However, although the influencing factors that contribute to the success of the implementation of green procurement are a crucial element of this policy, they have not yet been explored for the Greek public sector. Methods: This study applied data collection and a combined qualitative and quantitative data analysis. The research was divided into two phases: (i) the identification of critical factors (CFs) based on the literature, and (ii) an analysis of fourteen experts’ insights into those factors employing the Grey DEMATEL approach. Results: Based on the surveyed literature, fourteen CFs that contribute to the successful implementation of GPP were identified. From the analysis of the experts’ views, the factors were classified into two groups. Each group contains seven CFs. The CFs in the first group (causes) affect the CFs in the second group (effects). Conclusions: This study of the success factors in implementing green procurement in Greek public organizations can be further improved by incorporating new factors, as well as by utilizing the presented results in the follow-up of the NAP.