With the rapid growth and rising complexity of web content, edge-deployed LLMs have become essential for enhancing users' online experiences. However, sparsely-activated Mixture-of-Experts (MoE) models, which are well-suited for edge scenarios, face significant memory bottleneck challenges. Offload-based methods have been proposed to mitigate the problem, but they face difficulties with expert prediction. To promote the application of MoE models in edge scenarios, we propose Fate, an offloading system designed for MoE models to enable efficient inference in resource-constrained environments. The key insight behind Fate is that gate inputs from adjacent layers can be effectively used for expert prefetching, achieving high prediction accuracy. Furthermore, Fate employs a shallow-favoring expert caching strategy that increases the expert hit rate to 99%. Additionally, Fate integrates tailored quantization strategies for cache optimization and I/O efficiency. Experimental results show that, compared to baselines, Fate achieves up to 1.34×-5.07× prefill speedup and 1.26×-4.41× decoding speedup, while maintaining inference quality.