Assessing the reliability of deteriorating structures remains a challenging task. Traditional methods involve continuous reliability simulations at each time step, which leads to expensive computational expenses. To improve the computational efficiency, a Bayesian Sparse Grid (BSG) approach is proposed in this study. It effectively salvages information to estimate reliability analysis. The proposed BSG approach consists of the initial and operational stage. In the initial stage, integration samples of each dimension are generated. For the operational stage, these samples are dynamically weighted using Bayesian inference combined with the Smolyak algorithm at each time instant. By updating the weights and salvaging the generated samples, the proposed approach allows a cost-effective reconstruction of structural reliability without extra sampling. The efficacy of the proposed approach is demonstrated by numerical examples that encompass both explicit and implicit time-variant performance functions.