Powerful yet complex deep neural networks (DNNs) have fueled a booming demand for efficient DNN solutions to bring DNN-powered intelligence into numerous applications. Jointly optimizing the networks and their accelerators are promising in providing optimal performance. However, the great potential of such solutions have yet to be unleashed due to the challenge of simultaneously exploring the vast and entangled, yet different design spaces of the networks and their accelerators. To this end, we propose DNA, a Differentiable Network-Accelerator co-search framework for automatically searching for matched networks and accelerators to maximize both the task accuracy and acceleration efficiency. Specifically, DNA integrates two enablers, (1) a generic design space for DNN accelerators that is applicable to both FPGA- and ASIC-based DNN accelerators and compatible with DNN frameworks such as PyTorch to enable algorithmic exploration for more efficient DNNs and their accelerators; and (2) a joint DNN network and accelerator co-search algorithm that enables simultaneously searching for optimal DNN structures and their accelerators' micro-architectures and mapping methods to maximize both the task accuracy and acceleration efficiency. Experiments and ablation studies based on FPGA measurements and ASIC synthesis show that the matched networks and accelerators generated by DNA consistently outperform state-of-the-art (SOTA) DNNs and DNN accelerators (e.g., 3.04x better FPS with a 5.46% higher accuracy on ImageNet), while requiring notably reduced search time (up to 1234.3x) over SOTA co-exploration methods, when evaluated over ten SOTA baselines on three datasets. All codes will be released upon acceptance.