The monthly terrestrial water storage anomaly (TWSA) observations during the gap period between the Gravity Recovery and Climate Experiment (GRACE) satellite and its Follow-On (GRACE-FO) are missing, leading to discontinuity in the time series, and thus, impeding full utilization and analysis of the data. Despite previous efforts undertaken to tackle this issue, a gap-filling TWSA product with desirable accuracy at a global scale is still lacking. In this study, a straightforward and hydroclimatic data-driven Bayesian convolutional neural network (BCNN) is proposed to bridge this gap. Benefiting from the excellent capability of BCNN in handling image data and the integration of recent deep learning advances (including residual-skip connections and spatial-channel attentions), the proposed method can automatically extract informative features for TWSA predictions from multiple predictor data. The BCNN predictions are compared with reanalyzed/simulated TWSA, Swarm solution, and the TWSA prediction products generated by three recent studies, using commonly used accuracy metrics. Results demonstrate BCNN’s superior performance to obtain higher-quality TWSA predictions, particularly in relatively arid regions. Additionally, a comparison with two independent datasets at the basin scale further suggests that the BCNN-infilled TWSA is reliable to bridge the gap and enhance data consistency. Our gap-filling product can ultimately contribute to correcting the bias in long-term trend estimates, maintaining the continuity of TWSA time series and thus benefiting subsequent applications desiring continuous data records.