Systolic array has emerged as a prominent architecture for Deep Neural Network (DNN) hardware accelerators, providing high-throughput and low-latency performance essential for deploying DNNs across diverse applications. However, when used in safety-critical applications, reliability assessment is mandatory to guarantee the correct behavior of DNN accelerators. While fault injection stands out as a well-established practical and robust method for reliability assessment, it is still a very time-consuming process. This paper addresses the time efficiency issue by introducing a novel hierarchical software-based hardware-aware fault injection strategy tailored for systolic array-based DNN accelerators.