Training control policies in simulation is more appealing than on real robots directly, as it allows for exploring diverse states in an efficient manner. Yet, robot simulators inevitably exhibit disparities from the real-world \rebut{dynamics}, yielding inaccuracies that manifest as the dynamical simulation-to-reality (sim-to-real) gap. Existing literature has proposed to close this gap by actively modifying specific simulator parameters to align the simulated data with real-world observations. However, the set of tunable parameters is usually manually selected to reduce the search space in a case-by-case manner, which is hard to scale up for complex systems and requires extensive domain knowledge. To address the scalability issue and automate the parameter-tuning process, we introduce COMPASS, which aligns the simulator with the real world by discovering the causal relationship between the environment parameters and the sim-to-real gap. Concretely, our method learns a differentiable mapping from the environment parameters to the differences between simulated and real-world robot-object trajectories. This mapping is governed by a simultaneously learned causal graph to help prune the search space of parameters, provide better interpretability, and improve generalization on unseen parameters. We perform experiments to achieve both sim-to-sim and sim-to-real transfer, and show that our method has significant improvements in trajectory alignment and task success rate over strong baselines in several challenging manipulation tasks.