The energy cost minimization for mission-critical internet-of-things (IoT) in mobile edge computing (MEC) system is investigated in this work. Therein, short data packets are transmitted between the IoT devices and the access points (APs) to reduce transmission latency and prolong the battery life of the IoT devices. The effects of short-packet transmission on the radio resource allocation is explicitly revealed. We mathematically formulate the energy cost minimization problem as a mixed-integer non-linear programming (MINLP) problem, which is difficult to solve in an optimal way. More specifically, the difficulty is essentially derived from the coupling of the binary offloading variables and the resource management among all the IoT devices. For analytical tractability, we decouple the mixed-integer and non-convex optimization problem into two sub-problems, namely, the task offloading decision-making and the resource optimization problems, respectively. It is proved that the resource allocation problem for IoT devices under the fixed offloading strategy is convex. On this basis, an iterative algorithm is designed, whose performance is comparable to the best solution for exhaustive search, and aims to jointly optimize the offloading strategy and resource allocation. Simulation results verify the convergence performance and energy-saving function of the designed joint optimization algorithm. Compared with the extensive baselines under comprehensive parameter settings, the algorithm has better energy-saving effects.