Mixture of Experts (MoE) has become a mainstream architecture for building Large Language Models (LLMs) by reducing per-token computation while enabling model scaling. It can be viewed as partitioning a large Feed-Forward Network (FFN) at the tensor level into fine-grained sub-FFNs, or experts, and activating only a sparse subset for each input. While this sparsity improves efficiency, MoE still faces substantial challenges due to their massive computational scale and unpredictable activation patterns. To enable efficient MoE deployment, we identify dual sparsity at the tensor and neuron levels in pre-trained MoE modules as a key factor for both accuracy and efficiency. Unlike prior work that increases tensor-level sparsity through finer-grained expert design during pre-training, we introduce post-training expert partitioning to induce such sparsity without retraining. This preserves the mathematical consistency of model transformations and enhances both efficiency and accuracy in subsequent fine-tuning and inference. Building upon this, we propose DualSparse-MoE, an inference system that integrates dynamic tensor-level computation dropping with static neuron-level reconstruction to deliver significant efficiency gains with minimal accuracy loss. Experimental results show that enforcing an approximate 25% drop rate with our approach reduces average accuracy by only 0.08%-0.28% across three prevailing MoE models, while nearly all degrees of computation dropping consistently yield proportional computational speedups. Furthermore, incorporating load-imbalance awareness into expert parallelism achieves a 1.41x MoE module speedup with just 0.5% average accuracy degradation.