Intersection management with mixed cooperative and non-cooperative vehicles is crucial in next-generation transportation systems. This letter addresses the intersection scheduling problem in mixed systems by exploiting the benefit of cooperation based on the minimax scheduling framework, which was originally proposed for fully non-cooperative systems. Specifically, a long-horizon self-organization policy is first developed to optimize the passing order of cooperative vehicles in a distributed manner, which is proved convergent in a long enough system. Then a short-horizon trajectory planning policy is proposed to improve the efficiency when both cooperative and non-cooperative vehicles exist, and its safety and efficiency are theoretically validated. Furthermore, numerical simulations verify that the proposed policies can effectively reduce the scheduling cost and improve the throughput for cooperative vehicles.