Start-up accelerators are units supporting entrepreneurs (substantively, financially, legally, and organizationally) in establishing and running young and innovative companies such as start-ups. The commencing energy crisis has led to the need for energy savings, as well as the need to change energy policies and implement energy transformation, creating a wide field for start-ups and start-up accelerators. Making full use of potentially innovative solutions developed by start-ups is, in turn, essential for energy giants and related accelerators in the market. This has created the need for specific research in this direction. Therefore, in this paper we review the literature for 2011–2021 with respect to the role of accelerators supporting start-ups. A survey based on the Scopus database resulted in the identification of 76 papers on accelerators. Particular attention was paid to aspects of sustainable development (economic, social, and environmental). The analysis indicated that 75 works dealt with the economic aspects, 35 focused on social aspects, and only 5 focused on environmental aspects. Due to the perceived gap related to the small number of works dealing with environmental aspects, a decision was made to analyse the energy sector and the methods of operation of accelerators in this sector. Three models found in the market that were used by energy companies to cooperate with start-ups were characterized, along with their goals, in an attempt to enable an improved understanding as to which acceleration model best fits a manager’s organization. The models were supported by case studies—a model of commercial cooperation between a corporation and an external accelerator, a model of creating a corporate accelerator within an internal structure, and a hybrid model.