Coastal regions in North America face major threats from storm surges caused by hurricanes and nor'easters. Traditional numerical models, while accurate, are computationally expensive, limiting their practicality for real-time predictions. Recently, deep learning techniques have been developed for efficient simulation of time-dependent storm surge. To resolve the small scales of storm surge in both time and space over a long duration and a large area, these simulations typically need to employ oversized neural networks that struggle with the accumulation of prediction errors over successive time steps. To address these challenges, this study introduces a hierarchical deep neural network (HDNN) combined with a convolutional autoencoder (CAE) to accurately and efficiently predict storm surge time series. The CAE reduces the dimensionality of storm surge data, streamlining the learning process. HDNNs then map storm parameters to the low-dimensional representation of storm surge, allowing for sequential predictions across different time scales. Specifically, the current-level neural network is utilized to predict future states with a relatively large time step, which are passed as inputs to the next-level neural network for smaller time-step predictions. This process continues sequentially for all time steps. The results from different-level neural networks across various time steps are then stacked to acquire the entire time series of storm surge. The simulated low-dimensional representations are finally decoded back into storm surge time series. The proposed model was trained and tested using synthetic data from the North Atlantic Comprehensive Coastal Study. Results demonstrate its excellent performance to effectively handle high-dimensional surge data while mitigating the accumulation of prediction errors over time, making it a promising tool for advancing storm surge prediction.