Generative AI is an emerging tool in higher education; however, its connection with transversal competencies, as well as their sustainable adoption, remains underexplored. The study aims to analyze the scientific and conceptual development of generative artificial intelligence in higher education to identify the most relevant transversal competencies, strategic processes for its sustainable implementation, and global trends in academic production. A systematic literature review (PRISMA) was conducted on the Web of Science, Scopus, and PubMed, analyzing 35 studies for narrative synthesis and 897 publications for bibliometric analysis. The transversal competencies identified were: Academic Integrity, Critical Thinking, Innovation, Ethics, Creativity, Communication, Collaboration, AI Literacy, Responsibility, Digital Literacy, AI Ethics, Autonomous Learning, Self-Regulation, Flexibility, and Leadership. The conceptual framework connotes the interdisciplinary nature and five key processes were identified to achieve the sustainable integration of Generative AI in higher education oriented to the development of transversal competencies: (1) critical and ethical appropriation, (2) institutional management of technological infrastructure, (3) faculty development, (4) curricular transformation, and (5) pedagogical innovation. On bibliometric behavior, scientific articles predominate, with few systematic reviews. China leads in publication volume, and social sciences are the most prominent area. It is concluded that generative artificial intelligence is key to the development of transversal competencies if it is adopted from a critical, ethical, and pedagogically intentional approach. Its implications and future projections in the field of higher education are discussed.