Sustainability can be addressed in higher education (HE) in order to transform society. From this perspective, the curriculum of different subjects in both the sciences and the humanities allows for the inclusion of different Sustainable Development Goals (SDGs) promoted by the United Nations applied to analyze and to develop projects in the classroom measuring their respective social and environmental impacts. Thus, teaching in HE based on transformative learning implies that students can improve their future professional decisions using theoretical and practical knowledge with a social impact. The objective of the research presented here, based on the European Erasmus+ Project Social Impact of Higher Education on Sustainable Development, is to measure the effectiveness of the explanations focused on making higher education students aware of the basic concepts of the SDGs and to measure whether they can be deemed capable of developing solutions to social problems—thus becoming agents of change within their respective professional fields. The research method adopted here is a quantitative analysis, with a sample of 43 postgraduate students from different programmes in Spain. The results show that the students understand the different concepts of the SDGs dimensions and they can become agents of sustainable change in companies; however, it can also be concluded that they cannot measure economic consequences, due to the lack of a financial background of the students.