As the importance of the transition to sustainable development is increasingly recognised by individuals, organisations, and society as a whole, there is a growing need to examine its impact at micro-, meso-, and macro-levels. There is an urgent imperative to ensure the sustainability of growing economic inequalities, a degraded environment, and people living in uneven conditions in different societies. The authors, therefore, highlight the strategic role and essential contribution of organisations, and universities/higher education institutions in particular, in achieving sustainable development and the United Nations Sustainable Development Goals (SDGs). Universities/higher education institutions play a key role in fostering entrepreneurship and innovation, and they form the crucial architecture of contemporary practices in national economies and beyond. Policy makers, university/higher education institution governors, managers, and professors shape students and create new social contexts, and these must be oriented towards sustainability. This paper aims to explore the strategic role of organisations, in particular, universities/higher education institutions, as a key link between personal and social responsibility and, thus, as a powerful enabler of sustainable development. The authors examine the strategic transition to sustainability of two higher education institutions, the University of Maribor and the University of Economics—Varna, and conduct a qualitative case study research to develop a cybernetic model of the university’s/higher education institution’s transition to sustainability, which reflects the organisation’s growing commitment to achieving the Sustainability Development Goals. The model includes seven successive stages: pre-awareness, awareness, focusing, implementation, reaching out, transparency and disclosure, and continuous improvement. The study shows that sustainable development, i.e., sustainability governance, management, and operations, are indispensable for implementing the strategic concept of sustainability in an organisation and for achieving the strategic transition to sustainability as explained in the proposed cybernetic model.