The advent of artificial intelligence (AI) has precipitated a paradigm shift in healthcare; however, scholarly and policy discourse still prioritizes clinical applications, while managerial applications remain comparatively underexamined. This narrative review focuses on AI governance in hospital management and repositions AI as a managerial decision support infrastructure capable of reshaping hospital governance, organizational accountability, and institutional legitimacy. We conducted a structured narrative synthesis drawing on organizational theory, healthcare management, and AI governance literature. Relevant sources were identified through targeted database searches and citation tracking, screened for applicability to hospital-level managerial AI (strategic planning, resource allocation, and performance oversight), and synthesized via iterative thematic analysis to identify recurring governance challenges and convergent mechanisms. Based on this synthesis, we develop a three-dimensional governance framework linking managerial AI to (1) decision-authority distribution, (2) accountability mechanisms, and (3) institutional legitimacy. The framework is operationalized with illustrative scenarios and contextualized for the Turkish healthcare system. Our analysis shows that effective managerial AI governance requires explicit authority assignment, baseline thresholds for transparency and auditability, and board-level oversight structures to prevent responsibility diffusion and protect institutional legitimacy. We conclude with implementation guidance for hospital leaders and health system regulators.