Deep Neural Networks (DNNs) represent the state-of-the-art in many Artificial Intelligence (AI) tasks involving images, videos, text, and natural language. Their ubiquitous adoption is limited by the high computation and storage requirements of DNNs, especially for energy-constrained inference tasks at the edge using wearable and IoT devices. One promising approach to alleviate the computational challenges is implementing DNNs using low-precision fixed point (<16 bits) representation. However, the quantization error inherent in any Fixed Point (FxP) implementation limits the choice of bit-widths to maintain application-level accuracy. Prior e.orts recommend increasing the network size and/or re-training the DNN to minimize loss due to quantization, albeit with limited success. Complementary to the above approaches, we present Compensated-DNN, wherein we propose to dynamically compensate the error introduced due to quantization during execution. To this end, we introduce a new fixed-point representation viz. Fixed Point with Error Compensation (FPEC). The bits in FPEC are split between computation bits vs. compensation bits. The computation bits use conventional FxP notation to represent the number at low-precision. On the other hand, the compensation bits (1 or 2 bits at most) explicitly capture an estimate (direction and magnitude) of the quantization error in the representation. For a given word length, since FPEC uses fewer computation bits compared to FxP representation, we achieve a near-quadratic improvement in energy in the multiply-and-accumulate (MAC) operations. The compensation bits are simultaneously used by a low-overhead sparse compensation scheme to estimate the error accrued during MAC operations, which is then added to the MAC output to minimize the impact of quantization. We build compensated-DNNs for 7 popular image recognition benchmarks with 0.05-20.5 million neurons and 0.01-15.5 billion connections. Based on gate-level analysis at 14nm technology, we achieve 2.65.-4.88. and 1.13.-1.7. improvement in energy compared to 16-bit and 8-bit FxP implementations respectively, while maintaining <0.5% loss in classification accuracy.