This paper focuses on the finite element analysis simulation of the impact properties of composite sandwich structures made of carbon fiber-reinforced polymer lamina. In the existing studies, the composite sandwich structures with A-shaped cores have superior mechanical properties under quasi-static plane compression loads compared to W-shaped, Y-shaped, and X-shaped cores. However, there is limited research on the impact resistance of this structure. This paper studied the resistance of a composite A-shaped core structure to ballistic impact. Using ABAQUS/explicit finite element analysis software, ballistic impact tests for the composite A-shaped core structure were simulated based on the Hashin and Yeh failure criteria with a progressive damage model introduced in the user-defined subroutine VUMAT. First, the composite Y-shaped core sandwich structure was verified via experiments and simulations to determine the accuracy of the method, and then the composite A-shaped sandwich structure was subjected to a series of ballistic impact simulations. With varied impact velocity, the damage to the front and rear face sheet and cores via ballistic loads was simulated to illustrate the overall dynamic response process of the sandwich structure. Subsequently, a curve was fitted using a ballistic limit velocity equation, which was used as the criterion to evaluate the impact resistance of the composite A-shaped core structure. The results showed that, under the same relative density and the same number of component layers, the ballistic limit velocity of the composite A-shaped core sandwich structure was bigger than the composite Y-shaped core sandwich structure. The composite A-shaped core structure had 12.23% higher ballistic limit velocity than the composite Y-shaped core, indicating the impact resistance capabilities of the A-shaped core structure. In addition, the impact location’s effect on the impact response was investigated.