The effectiveness of deep neural networks (DNN) in vision, speech, and language processing has prompted a tremendous demand for energy-efficient high-performance DNN inference systems. Due to the increasing memory intensity of most DNN workloads, main memory can dominate the system's energy consumption and stall time. One effective way to reduce the energy consumption and increase the performance of DNN inference systems is by using approximate memory, which operates with reduced supply voltage and reduced access latency parameters that violate standard specifications. Using approximate memory reduces reliability, leading to higher bit error rates. Fortunately, neural networks have an intrinsic capacity to tolerate increased bit errors. This can enable energy-efficient and high-performance neural network inference using approximate DRAM devices. Based on this observation, we propose EDEN, a general framework that reduces DNN energy consumption and DNN evaluation latency by using approximate DRAM devices, while strictly meeting a user-specified target DNN accuracy. EDEN relies on two key ideas: 1) retraining the DNN for a target approximate DRAM device to increase the DNN's error tolerance, and 2) efficient mapping of the error tolerance of each individual DNN data type to a corresponding approximate DRAM partition in a way that meets the user-specified DNN accuracy requirements. We evaluate EDEN on multi-core CPUs, GPUs, and DNN accelerators with error models obtained from real approximate DRAM devices. For a target accuracy within 1% of the original DNN, our results show that EDEN enables 1) an average DRAM energy reduction of 21%, 37%, 31%, and 32% in CPU, GPU, and two DNN accelerator architectures, respectively, across a variety of DNNs, and 2) an average (maximum) speedup of 8% (17%) and 2.7% (5.5%) in CPU and GPU architectures, respectively, when evaluating latency-bound DNNs.