The accurate and timely diagnosis of sensor faults plays a critical role in ensuring the reliability and performance of structural health monitoring (SHM) systems. However, the challenge is detecting, locating, and estimating sensor faults in an online manner using limited training data. To resolve this problem, a novel approach for online sensor fault diagnosis is proposed for SHM. The proposed approach is based on meta-learning, which enables superior model generalization capabilities using limited data. The detection, localization, and estimation of typical sensor faults in an online manner can be achieved efficiently by the proposed approach. First, a one-dimensional convolutional neural network (1D CNN) is designed to detect and locate faulty sensors. The initial model parameters of the 1D CNN are optimized using a model-agnostic meta-learning training strategy. This strategy allows the acquisition of transferable prior knowledge, which can speed up the learning process on new sensor fault detection and localization tasks. The meta-learning strategy also enables efficient and accurate detection and localization of potential faulty sensors with limited data. After detecting and locating the faulty sensors, an online updating algorithm based on a dual Kalman filter is used to estimate the severity of sensor faults and structural states simultaneously. The proposed approach is demonstrated with simulated sensor faults that cover a numerical example and field measurements from the Canton Tower. The results show that the proposed approach is applicable for online sensor fault diagnosis in SHM.