The identification of gravitationally lensed supernovae in modern astronomical datasets is a needle-in-a-haystack problem with dramatic scientific implications: discovered systems can be used to directly measure and resolve the current tension on the value of the expansion rate of the Universe today. We hypothesize that the image-based features of the gravitational lensing and the temporal-based features of the time-varying brightness are equally important in classifications. We therefore develop a deep learning technique that utilizes long short-term memory cells for the time-varying brightness of astronomical systems and convolutional layers for the raw images of astronomical systems simultaneously, and then concatenates the feature maps with multiple fully connected layers. This novel approach achieves a receiver operating characteristic area under curve of 0.97 on simulated astronomical data and more importantly outperforms standalone versions of its recurrent and convolutional constituents. We find that combining recurrent and convolutional layers within one coherent network architecture allows the network to optimally weight and aggregate the temporal and image features to yield a promising tool for lensed supernovae identification.