Evaluation of tail probabilities in reliability analysis faces challenges in cases when complex models with high-dimensional random input are considered. To address such problems, we herein propose the use of surrogate models developed with low-rank tensor approximations. In this approach, the response quantity of interest is expressed as a sum of a few rank-one tensors. We first describe a non-intrusive method for building low-rank approximations using a greedy algorithm, which relies on the solution of minimization problems of only small size. In the sequel, we demonstrate the efficiency of meta-models built in this way in reliability applications involving the deflections of structural systems under static loads and the temperature in stationary heat conduction with spatially varying diffusion coefficient. Furthermore, we show that the proposed approach outperforms the popular meta-modeling technique of polynomial chaos expansions.