Learning analytics have proved promising capabilities and opportunities to many aspects of academic research and higher education studies. Data-driven insights can significantly contribute to provide solutions for curbing costs and improving education quality. This paper adopts a two-phase machine learning approach, which utilizes both unsupervised and supervised learning techniques for predicting outcomes of students following Higher Education programs of studies. The approach has been applied in a case-study which has been performed in the context of an undergraduate Computer Science curriculum offered by the University of Thessaly in Greece. Students involved in the case study were initially grouped based on the similarity of specific education-related factors and metrics. Using the K-Means algorithm, our clustering experiments revealed the presence of three coherent clusters of students. Subsequently, the discovered clusters were utilized to train prediction models for addressing each particular cluster of students individually. In this regard, two machine learning models were trained for every cluster of students in order to predict the time to degree completion and student enrollment in the offered educational programs. The developed models are claimed to produce predictions with relatively high accuracy. Finally, the paper discusses the potential usefulness of the clustering-aided approach for learning analytics in Higher Education.