Engineering structures, such as bridges and buildings, are impacted by uncertainties. Risk management, therefore, necessitates the prediction of failure probability to assess the structure's safety in the presence of uncertainties. Several methods, namely Monte Carlo Simulation (MCS), First-order Reliability Method (FORM), and Second-order Reliability Method (SORM), have been proposed to assess structural reliability. Yet, these methods encounter difficulties in dealing with high-dimensional problems as the required computation time grows exponentially. This study develops an integrated framework, Dimension-reduction Reliability Analysis (DRRA), to provide a remedy. To assess the structural reliability, DRRA adopts a deep learning technique to reduce the dimensionality of input variables for the active-learning reliability method AK-MCS. DRRA involves training an autoencoder and a deep feedforward network to map the training samples from a high-dimensional space to a lower-dimensional latent representation. Then, AK-MCS is employed to estimate structural reliability in the latent space. DRRA is demonstrated by modeling a practical high-dimensional case. The performance is compared with traditional AK-MCS (without dimensionality reduction) and another existing method: HDDA-GP. The comparison results confirm that the DRRA framework achieves better accuracy and higher efficiency than previous work. The application of the DRRA framework across various design cases will further demonstrate its overall performance.