Against the backdrop of the widespread integration of generative artificial intelligence (GAI) into higher education, previous research has yielded divergent findings regarding its relationship with students’ higher-order thinking (HOT). Moving beyond outcome-oriented discussions of technology use itself, this study examines the association between HOT performance, instructional design elements, and cognitive processes in GAI-supported teaching contexts from an instructional design perspective. A multimethod analysis combining structural equation modeling (SEM) and fuzzy-set qualitative comparative analysis (fsQCA) was conducted. The findings indicate that HOT performance cannot be explained by technology use alone. Rather, it is more likely to be associated with cognitive conflict and cognitive elaboration processes shaped by instructional design. Different instructional design elements played differentiated roles in triggering cognitive conflict and supporting cognitive elaboration. The overall influence of disciplinary context on these cognitive processes was limited, although one individual path showed a significant difference. The configurational analysis further showed that high HOT performance did not correspond to a single pathway but was associated with multiple combinations of instructional design elements and cognitive processes, demonstrating configurational multiplicity and asymmetry.