Deep neural networks (DNNs) have revolutionized tasks such as image classification and speech recognition but often falter when training and test data diverge in distribution. External factors, from weather effects on images to varied speech environments, can cause this discrepancy, compromising DNN performance. Online test-time adaptation (OTTA) methods present a promising solution, recalibrating models in real-time during the test stage without requiring historical data. However, the OTTA paradigm is imperfect, often falling prey to issues such as catastrophic forgetting due to its reliance on noisy, self-trained predictions. Although some contemporary strategies mitigate this by tying adaptations to the static source model, this restricts model flexibility. This paper introduces a continual momentum filtering (CMF) framework, leveraging the Kalman filter (KF) to strike a balance between model adaptability and information retention. The CMF intertwines optimization via stochastic gradient descent with a KF-based inference process. This methodology not only aids in averting catastrophic forgetting but also provides high adaptability to shifting data distributions. We validate our framework on various OTTA scenarios and real-world situations regarding covariate and label shifts, and the CMF consistently shows superior performance compared to state-of-the-art methods.