Performing reliability analysis on complex systems is often computationally expensive. In particular, when dealing with systems having high input dimensionality, reliability estimation becomes a daunting task. A popular approach to overcome the problem associated with time-consuming and expensive evaluations is building a surrogate model. However, these computationally efficient models often suffer from the curse of dimensionality. Hence, training a surrogate model for high-dimensional problems is not straightforward. Henceforth, this paper presents a framework for solving high-dimensional reliability analysis problems. The basic premise is to train the surrogate model on a low-dimensional manifold, discovered using the active subspace algorithm. However, learning the low-dimensional manifold using active subspace is non-trivial as it requires information on the gradient of the response variable. To address this issue, we propose using sparse learning algorithms in conjunction with the active subspace algorithm; the resulting algorithm is referred to as the sparse active subspace (SAS) algorithm. We project the high-dimensional inputs onto the identified low-dimensional manifold identified using SAS. A high-fidelity surrogate model is used to map the inputs on the low-dimensional manifolds to the output response. We illustrate the efficacy of the proposed framework by using three benchmark reliability analysis problems from the literature. The results obtained indicate the accuracy and efficiency of the proposed approach compared to already established reliability analysis methods in the literature.