Computing the cumulative failure probability for a given period of time is the main goal of a time-dependent reliability analysis. Estimating this probability is challenging for problems with nonmonotonic performance functions, especially when they are costly to evaluate and have high dimensionality. Discretizing the time interval is one main step in most of the time-dependent reliability methods. Hence, the problem can be converted into a serially connected system reliability problem. Therefore, efficient system reliability methods can be used for time-dependent reliability analysis. AK-SYS (Active learning and Kriging-based SYStem reliability method) is a Kriging-based method for system reliability assessment, including an active learning procedure for the enrichment process. In this paper, we exploit the efficiency of AK-SYS to propose a new time-dependent reliability method that is called AK-SYS-t. Two examples are used to compare the efficiency of the proposed method with competing methods, and a third example is used to highlight the opportunities offered by this method for fatigue reliability analysis. In the end, a crude approach is also proposed to provide the full curve of the cumulative failure probability.