In light of the drastically-increasing demands on ubiquitous information acquisition and exchange, battery-powered devices are playing a more critical role in wireless networking, which inspires and expedites the development of energy-harvesting techniques. The sustainability requirements for such devices include two mutual-impacting dimensions: energy supply avoiding battery outage and stable transmissions avoiding buffer overflow. We in this paper propose a power-and-rate adaption scheme subject to the constraints on buffer-overflow probability and battery-outage probability in energy-harvesting enabled networks. The two probability constraints address the statistical assurance for the two-dimensional sustainability, which is suitable to dealing with the highly unpredictable energy-harvesting and channel fading statuses. Specifically, by applying the asymptotic queuing analyses to data-transmission and energy-harvesting processes, we can characterize both of the battery-outage and buffer-overflow probabilities via the effective-capacity/-bandwidth theories. Then, we formulate the effective-capacity optimization problem, which maximizes the sustainable throughput while complying with the statistical buffer-overflow and battery-outage constraints, and solve for the optimal power-adaptation scheme. We also conducted abundant simulations to evaluate our scheme's performances, demonstrate the superiority of our proposed scheme over existing baseline schemes, as well as study the impact of sustainability requirements on the performances.