The time-variant reliability analysis method based on the adaptive single-loop surrogate has attracted much attention due to its excellent computing performance. However, the existing methods do not sufficiently focus surrogate learning on the reliability-sensitive space-time region with high efficacy in improving the reliability surrogate, resulting in calculating waste. In this paper, a reliability-sensitive space-time Kriging (RSTK) modeling approach is proposed. In the RSTK, to screen out reliability-sensitive trajectory segments, a reliability-sensitive space-time determination method is first proposed; further, to capture high-quality training samples, a reliability-sensitive space-time learning approach is designed correspondingly; finally, a matching iteration termination criterion is constructed. Four case studies demonstrate the superiority of the proposed RSTK in reducing calculational costs. RSTK shortens the iteration time by one to two orders of magnitude and reduces the surrogate cost by up to 22.3% while maintaining accuracy.