This paper investigates a self-organized critical approach for dynamically load-balancing computational workloads. The proposed model is based on the Bak-Tang-Wiesenfeld sandpile: a cellular automaton that works in a critical regime at the edge of chaos. In analogy to grains of sand, tasks arrive and pile up on the different processing elements or sites of the system. When a pile exceeds a certain threshold, it collapses and initiates an avalanche of migrating tasks, i. e., producing load-balancing. We show that the frequency of such avalanches is in power-law relation with their sizes, a scale-invariant fingerprint of self-organized criticality that emerges without any tuning of parameters. Such an emergent pattern has organic properties such as the self-organization of tasks into resources or the self-optimization of the computing performance. The conducted experimentation also reveals that the system has a critical attractor in the point in which the arrival rate of tasks equals the processing power of the system. Taking advantage of this fact, we hypothesize that the processing elements can be turned on and off depending on the state of the workload as to maximize the utilization of resources. An interesting side effect is that the overall energy consumption of the system is minimized without compromising the quality of service.