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In this paper, we co-design the transmission power and the offloading strategy for job offloading to a mobile edge computing(MEC) server at Terahertz (THz) frequencies.The goal is to minimize the communication energy consumption while providing ultra-re-liable low end-to-end latency (URLLC) ser-vices. To that end, we first establish a novel reliability framework, where the end-to-end(E2E) delay equals a weighted sum of the local computing delay, the communication delay and the edge computing delay, and the reliability is defined as the probability that the E2E delay remains below a certain pre-defined threshold. This reliability gives a full view of the statistics of the E2E delay, thus constitut-ing advancement over prior works that have considered only average delays. Based on this framework, we establish the communication energy consumption minimization problem under URLLC constraints. This optimization problem is non-convex. To handle that is-sue, we first consider the special single-user case, where we derive the optimal solution by analyzing the structure of the optimization problem. Further, based on the analytical re-sult for the single-user case, we decouple the optimization problem for multi-user scenarios into several sub-optimization problems and propose a sub-optimal algorithm to solve it.Numerical results verify the performance of the proposed algorithm.