Mission-critical wireless sensor networks are attractive for information gathering in various complex environments and support many mission-critical applications, such as industrial automation and security surveillance. However, in order to fully exploit these networks for such applications, agile and scalable network management for data transfer and computing task implementation are essential. Thus, in this paper, we propose a software-defined mission-critical wireless sensor network (MC-SDWSN) which can solve the existing challenging issues in tradition WSNs, such as resource utilization, data processing, system compatibility, and strict latency requirement. The architecture is based on the idea of SDN architecture, combining the hierarchical cloud and edge computing technologies. Based on the MC-SDWSN architecture, a novel centralized computation offload strategy in sensor network application is proposed to show the feasibility. The simulation results in confirm the MC-SDWSN architecture, and the edge offloading strategy could support the critical missions effectively.