On-device learning is an emerging technique to pave the last mile of enabling edge intelligence, which eliminates the limitations of conventional in-cloud computing where dozens of computational capacities and memories are needed. A high-performance on-device learning system requires breaking the constraints of limited resources and alleviating computational overhead. In this paper, we show that employing the 8-bit fixed-point (INT8) quantization in both forward and backward passes over a deep model is a promising way to enable tiny on-device learning in practice. The key to an efficient quantization-aware training method is to exploit the hardware-level enabled acceleration while preserving the training quality in each layer. However, off-the-shelf quantization methods cannot handle the on-device learning paradigm of fixed-point processing. To overcome these challenges, we propose a novel INT8 training method, which optimizes the computation of forward and backward passes via the delicately designed Loss-aware Compensation (LAC) and Parameterized Range Clipping (PRC), respectively. Specifically, we build a new network component, the compensation layer, to automatically counteract the quantization error of tensor arithmetic. We implement our method in Octo, a lightweight cross-platform system for tiny on-device learning. Evaluation on commercial AI chips shows that Octo holds higher training efficiency over state-of-the-art quantization training methods, while achieving adequate processing speedup and memory reduction over the full-precision training.