This paper presents a systematic review of the role of prompt engineering during interactions with Generative Artificial Intelligence (GenAI) in Higher Education (HE) to discover potential methods of improving educational outcomes. Drawing on a comprehensive search of academic databases and relevant literature, key trends, including multiple framework designs, are presented and explored to review the role, relevance, and applicability of prompt engineering to purposefully improve GenAI-generated responses in higher education contexts. Multiple experiments using a variety of prompt engineering frameworks are compared, contrasted and discussed. Analysis reveals that well-designed prompts have the potential to transform interactions with GenAI in higher education teaching and learning. Further findings show it is important to develop and teach pragmatic skills in AI interaction, including meaningful prompt engineering, which is best managed through a well-designed framework for creating and evaluating GenAI applications that are aligned with pre-determined contextual educational goals. The paper outlines some of the key concepts and frameworks that educators should be aware of when incorporating GenAI and prompt engineering into their teaching practices, and when teaching students the necessary skills for successful GenAI interaction.