Creativity is widely recognized as a critical competency for addressing the complex challenges of the twenty-first century. At the same time, Generative Artificial Intelligence (GenAI) is transforming how students learn, think, and solve problems in higher education. In human-AI collaboration, this study first identified students’ cognitive strategies in GenAI-supported contexts, and examined the relationship between the cognitive strategy used and the creativity (i.e., originality and quality) of each idea. Second, this study identified latent profiles of students’ cognitive strategy use, and their relationship with creativity. Using a novel process-oriented theoretical framework and a combination of deductive and inductive analyses, six cognitive strategies were identified: repeat, unmediated, experiential, conceptual combination, semantic, and assumption reversal. A laboratory-based observational study involving 146 undergraduate students showed that constructive cognitive strategies (conceptual combination and semantic) were more frequently associated with highly original ideas than active cognitive strategies (unmediated and experiential), whereas no differences were found for idea quality. Furthermore, latent profile analysis identified four distinct profiles of human–AI collaboration: Largely Independent Thinkers, AI-Constructive Combiners, AI-Balanced Thinkers, and AI-Reliant Reproducers. Although the profiles did not differ in their average originality, they differed significantly in their average quality. Students classified as AI-Constructive Combiners, AI-Balanced Thinkers, and AI-Reliant Reproducers produced ideas of higher quality than the Largely Independent Thinkers. These findings demonstrate that the way students cognitively engage with GenAI shapes their creativity and provide insights for fostering effective human–AI collaboration in higher education.