The increasing adoption of approximate computing in deep neural network accelerators (AxDNNs) promises significant energy efficiency gains. However, permanent faults in AxDNNs can severely degrade their performance compared to their accurate counterparts (AccDNNs). Traditional fault detection and mitigation approaches, while effective for AccDNNs, introduce substantial overhead and latency, making them impractical for energy-constrained real-time deployment. To address this, we introduce EPSILON, a lightweight framework that leverages pre-computed statistical signatures and layer-wise importance metrics for efficient fault detection and mitigation in AxDNNs. Our framework introduces a novel non-parametric pattern-matching algorithm that enables constant-time fault detection without interrupting normal execution while dynamically adapting to different network architectures and fault patterns. EPSILON maintains model accuracy by intelligently adjusting mitigation strategies based on a statistical analysis of weight distribution and layer criticality while preserving the energy benefits of approximate computing. Extensive evaluations across various approximate multipliers, AxDNN architectures, popular datasets (MNIST, CIFAR-10, CIFAR-100, ImageNet-1k), and fault scenarios demonstrate that EPSILON maintains 80.05\% accuracy while offering 22\% improvement in inference time and 28\% improvement in energy efficiency, establishing EPSILON as a practical solution for deploying reliable AxDNNs in safety-critical edge applications.